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Gráfico de la función y =:
x*e^(-((x^2)/2))
(x+2)/(x-4)
(x^2+8)/(x+1)
(x+2)^(2/3)-(x-2)^(2/3)
Expresiones idénticas
y=(x- dos)x^ dos -5x+ cuatro /(x- cuatro)
y es igual a (x menos 2)x al cuadrado menos 5x más 4 dividir por (x menos 4)
y es igual a (x menos dos)x en el grado dos menos 5x más cuatro dividir por (x menos cuatro)
y=(x-2)x2-5x+4/(x-4)
y=x-2x2-5x+4/x-4
y=(x-2)x²-5x+4/(x-4)
y=(x-2)x en el grado 2-5x+4/(x-4)
y=x-2x^2-5x+4/x-4
y=(x-2)x^2-5x+4 dividir por (x-4)
Expresiones semejantes
y=(x-2)x^2+5x+4/(x-4)
y=(x+2)x^2-5x+4/(x-4)
y=(x-2)x^2-5x-4/(x-4)
y=(x-2)x^2-5x+4/(x+4)

Trazado el gráfico de la función/ y=(x-2)x^2-5x+4/(x-4)

Gráfico de la función y = y=(x-2)x^2-5x+4/(x-4)

Solución

Ha introducido [src]

                2           4  
f(x) = (x - 2)*x  - 5*x + -----
                          x - 4

f{\left(x \right)} = \left(x^{2} \left(x - 2\right) - 5 x\right) + \frac{4}{x - 4}

f = x^2*(x - 2) - 5*x + 4/(x - 4)

Gráfico de la función

Dominio de definición de la función

Puntos en los que la función no está definida exactamente:

x_{1} = 4

Puntos de cruce con el eje de coordenadas X

El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:

\left(x^{2} \left(x - 2\right) - 5 x\right) + \frac{4}{x - 4} = 0

Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica

x_{1} = - \frac{\sqrt{\frac{139}{6 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}}} + 7 + 2 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}} - \frac{139}{6 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}}} + \frac{4}{\sqrt{\frac{139}{6 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}}} + 7 + 2 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}}}} + 14}}{2} + \frac{3}{2}

x_{2} = - \frac{\sqrt{\frac{139}{6 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}}} + 7 + 2 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}} - \frac{139}{6 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}}} + \frac{4}{\sqrt{\frac{139}{6 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}}} + 7 + 2 \sqrt[3]{\frac{\sqrt{1293}}{9} + \frac{317}{8}}}} + 14}}{2} + \frac{3}{2}

Solución numérica

x_{1} = -0.209431189721467

x_{2} = -0.209431189721563

x_{3} = -1.3316546931043

Puntos de cruce con el eje de coordenadas Y

El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en (x - 2)*x^2 - 5*x + 4/(x - 4).

\frac{4}{-4} + \left(- 2 \cdot 0^{2} - 0\right)

Resultado:

f{\left(0 \right)} = -1

Punto:

(0, -1)

Extremos de la función

Para hallar los extremos hay que resolver la ecuación

\frac{d}{d x} f{\left(x \right)} = 0

(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:

\frac{d}{d x} f{\left(x \right)} =

primera derivada

x^{2} + 2 x \left(x - 2\right) - 5 - \frac{4}{\left(x - 4\right)^{2}} = 0

Resolvermos esta ecuación
Raíces de esta ecuación

x_{1} = \frac{7}{3} - \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} - \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} + \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}

x_{2} = \frac{7}{3} - \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} + \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} + \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}

x_{3} = \frac{7}{3} + \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} - \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} - \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}

x_{4} = \frac{7}{3} + \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} - \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2} + \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2}

Signos de extremos en los puntos:

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                                                                                                                                                                                                                                                          /                                                                                /                                              /  361   I*\/ 9019          /                                              /  361   I*\/ 9019                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
                                                                                                                                                                                                                                                         /                                                                                /                                         18*3 /   --- + ----------        /                                         18*3 /   --- + ----------                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                   5   \/                                                                               \/                                             \/    216       54          \/                                             \/    216       54                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                 - - + --------------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                   3                                                                               2                                                                                                                  2

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2                                                                                                                                                                                                                                           
                                                                                              __________________________________________________________________________________________________________________________________________                                                                                                                                                                                                                                                                                                                                                  __________________________________________________________________________________________________________________________________________   /                                                                                             __________________________________________________________________________________________________________________________________________\  /                                                                                             __________________________________________________________________________________________________________________________________________\ 
                                                                                             /             __________________                                                                                                                                                                                                                                                                                                                                                                                                                                                            /             __________________                                                                                                              |                                                                                            /             __________________                                                                                                           |  |                                                                                            /             __________________                                                                                                           | 
                                                                                            /             /           ______                                                                                                                                                                                                                                                                                                                                                                                                                                                            /             /           ______                                                                                                               |                                                                                           /             /           ______                                                                                                            |  |                                                                                           /             /           ______                                                                                                            | 
                                                                                           /   92        /  361   I*\/ 9019                                        380                                                   65                                                                                                                                                                                                                                                                                                                                                            /   92        /  361   I*\/ 9019                                        380                                                   65                |                                                                                          /   92        /  361   I*\/ 9019                                        380                                                   65             |  |                                                                                          /   92        /  361   I*\/ 9019                                        380                                                   65             | 
                                                                                          /    -- - 2*3 /   --- + ----------  - -------------------------------------------------------------------------- - --------------------------                                                                                                                                                                                                                                                                                                                                  5*           /    -- - 2*3 /   --- + ----------  - -------------------------------------------------------------------------- - --------------------------    |                                                                                         /    -- - 2*3 /   --- + ----------  - -------------------------------------------------------------------------- - -------------------------- |  |                                                                                         /    -- - 2*3 /   --- + ----------  - -------------------------------------------------------------------------- - -------------------------- | 
               _____________________________________________________________             /     9      \/    216       54                     _____________________________________________________________           __________________                                                                                                                                                                                                                                                                  _____________________________________________________________               /     9      \/    216       54                     _____________________________________________________________           __________________    |              _____________________________________________________________             /     9      \/    216       54                     _____________________________________________________________           __________________ |  |              _____________________________________________________________             /     9      \/    216       54                     _____________________________________________________________           __________________ | 
              /             __________________                                          /                                                   /             __________________                                        /           ______                                                                                                                                                                                                                                                                  /             __________________                                            /                                                   /             __________________                                        /           ______     |             /             __________________                                          /                                                   /             __________________                                        /           ______  |  |             /             __________________                                          /                                                   /             __________________                                        /           ______  | 
             /             /           ______                                          /                                                   /             /           ______                                        /  361   I*\/ 9019                                                                                                                                                                                                                                                                  /             /           ______                                            /                                                   /             /           ______                                        /  361   I*\/ 9019      |            /             /           ______                                          /                                                   /             /           ______                                        /  361   I*\/ 9019   |  |            /             /           ______                                          /                                                   /             /           ______                                        /  361   I*\/ 9019   | 
            /   46        /  361   I*\/ 9019                 65                       /                                                   /   46        /  361   I*\/ 9019                 65                18*3 /   --- + ----------                                                                                                                                                                                                                                                                /   46        /  361   I*\/ 9019                 65                         /                                                   /   46        /  361   I*\/ 9019                 65                18*3 /   --- + ----------     |           /   46        /  361   I*\/ 9019                 65                       /                                                   /   46        /  361   I*\/ 9019                 65                18*3 /   --- + ----------  |  |           /   46        /  361   I*\/ 9019                 65                       /                                                   /   46        /  361   I*\/ 9019                 65                18*3 /   --- + ----------  | 
           /    -- + 2*3 /   --- + ----------  + --------------------------          /                                          27*      /    -- + 2*3 /   --- + ----------  + --------------------------       \/    216       54                                                                                                                                                                                                                                                           5*      /    -- + 2*3 /   --- + ----------  + --------------------------            /                                          27*      /    -- + 2*3 /   --- + ----------  + --------------------------       \/    216       54         |          /    -- + 2*3 /   --- + ----------  + --------------------------          /                                          27*      /    -- + 2*3 /   --- + ----------  + --------------------------       \/    216       54      |  |          /    -- + 2*3 /   --- + ----------  + --------------------------          /                                          27*      /    -- + 2*3 /   --- + ----------  + --------------------------       \/    216       54      | 
          /     9      \/    216       54                __________________         /                                                   /     9      \/    216       54                __________________                                                                                                                                                                                                                                                                                           /     9      \/    216       54                __________________           /                                                   /     9      \/    216       54                __________________                                  |         /     9      \/    216       54                __________________         /                                                   /     9      \/    216       54                __________________                               |  |         /     9      \/    216       54                __________________         /                                                   /     9      \/    216       54                __________________                               | 
         /                                              /           ______         /                                                   /                                              /           ______                                                                                                                                                                                                                                                                                           /                                              /           ______           /                                                   /                                              /           ______                                   |        /                                              /           ______         /                                                   /                                              /           ______                                |  |        /                                              /           ______         /                                                   /                                              /           ______                                | 
        /                                              /  361   I*\/ 9019         /                                                   /                                              /  361   I*\/ 9019                                                                                                                                                                                                                                                                                           /                                              /  361   I*\/ 9019           /                                                   /                                              /  361   I*\/ 9019                                    |       /                                              /  361   I*\/ 9019         /                                                   /                                              /  361   I*\/ 9019                                 |  |       /                                              /  361   I*\/ 9019         /                                                   /                                              /  361   I*\/ 9019                                 | 
       /                                         18*3 /   --- + ----------       /                                                   /                                         18*3 /   --- + ----------                                                                                                                                                                                                                                                                                         /                                         18*3 /   --- + ----------         /                                                   /                                         18*3 /   --- + ----------                                   |      /                                         18*3 /   --- + ----------       /                                                   /                                         18*3 /   --- + ----------                                |  |      /                                         18*3 /   --- + ----------       /                                                   /                                         18*3 /   --- + ----------                                | 
 7   \/                                             \/    216       54         \/                                                  \/                                             \/    216       54                                        35                                                                                                                       4                                                                                                                         \/                                             \/    216       54           \/                                                  \/                                             \/    216       54                                       |7   \/                                             \/    216       54         \/                                                  \/                                             \/    216       54                                    |  |1   \/                                             \/    216       54         \/                                                  \/                                             \/    216       54                                    | 
(- + ----------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------, - -- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------- + |- + ----------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------| *|- + ----------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------|)
 3                                      2                                                                                                                  2                                                                                3                                                                                                   __________________________________________________________________________________________________________________________________________                                       2                                                                                                                    2                                                                                \3                                      2                                                                                                                  2                                                                            /  \3                                      2                                                                                                                  2                                                                            / 
                                                                                                                                                                                                                                                                                                                                               /             __________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
                                                                                                                                                                                                                                                                                                                                              /             /           ______                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                             /   92        /  361   I*\/ 9019                                        380                                                   65                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
                                                                                                                                                                                                                                                                                                                                            /    -- - 2*3 /   --- + ----------  - -------------------------------------------------------------------------- - --------------------------                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                                                 _____________________________________________________________             /     9      \/    216       54                     _____________________________________________________________           __________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                                                /             __________________                                          /                                                   /             __________________                                        /           ______                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                               /             /           ______                                          /                                                   /             /           ______                                        /  361   I*\/ 9019                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
                                                                                                                                                                                                                                                              /   46        /  361   I*\/ 9019                 65                       /                                                   /   46        /  361   I*\/ 9019                 65                18*3 /   --- + ----------                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                             /    -- + 2*3 /   --- + ----------  + --------------------------          /                                          27*      /    -- + 2*3 /   --- + ----------  + --------------------------       \/    216       54                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                            /     9      \/    216       54                __________________         /                                                   /     9      \/    216       54                __________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                           /                                              /           ______         /                                                   /                                              /           ______                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                          /                                              /  361   I*\/ 9019         /                                                   /                                              /  361   I*\/ 9019                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                                         /                                         18*3 /   --- + ----------       /                                                   /                                         18*3 /   --- + ----------                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                   5   \/                                             \/    216       54         \/                                                  \/                                             \/    216       54                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                 - - + ----------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                   3                                      2                                                                                                                  2

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2                                                                                                                                                                                                                                           
                                                                                              __________________________________________________________________________________________________________________________________________                                                                                                                                                                                                                                                                                                                                                  __________________________________________________________________________________________________________________________________________   /                                                                                             __________________________________________________________________________________________________________________________________________\  /                                                                                             __________________________________________________________________________________________________________________________________________\ 
                                                                                             /             __________________                                                                                                                                                                                                                                                                                                                                                                                                                                                            /             __________________                                                                                                              |                                                                                            /             __________________                                                                                                           |  |                                                                                            /             __________________                                                                                                           | 
                                                                                            /             /           ______                                                                                                                                                                                                                                                                                                                                                                                                                                                            /             /           ______                                                                                                               |                                                                                           /             /           ______                                                                                                            |  |                                                                                           /             /           ______                                                                                                            | 
                                                                                           /   92        /  361   I*\/ 9019                                        380                                                   65                                                                                                                                                                                                                                                                                                                                                            /   92        /  361   I*\/ 9019                                        380                                                   65                |                                                                                          /   92        /  361   I*\/ 9019                                        380                                                   65             |  |                                                                                          /   92        /  361   I*\/ 9019                                        380                                                   65             | 
                                                                                          /    -- - 2*3 /   --- + ----------  - -------------------------------------------------------------------------- - --------------------------                                                                                                                                                                                                                                                                                                                                  5*           /    -- - 2*3 /   --- + ----------  - -------------------------------------------------------------------------- - --------------------------    |                                                                                         /    -- - 2*3 /   --- + ----------  - -------------------------------------------------------------------------- - -------------------------- |  |                                                                                         /    -- - 2*3 /   --- + ----------  - -------------------------------------------------------------------------- - -------------------------- | 
               _____________________________________________________________             /     9      \/    216       54                     _____________________________________________________________           __________________                                                                                                                                                                                                                                                                  _____________________________________________________________               /     9      \/    216       54                     _____________________________________________________________           __________________    |              _____________________________________________________________             /     9      \/    216       54                     _____________________________________________________________           __________________ |  |              _____________________________________________________________             /     9      \/    216       54                     _____________________________________________________________           __________________ | 
              /             __________________                                          /                                                   /             __________________                                        /           ______                                                                                                                                                                                                                                                                  /             __________________                                            /                                                   /             __________________                                        /           ______     |             /             __________________                                          /                                                   /             __________________                                        /           ______  |  |             /             __________________                                          /                                                   /             __________________                                        /           ______  | 
             /             /           ______                                          /                                                   /             /           ______                                        /  361   I*\/ 9019                                                                                                                                                                                                                                                                  /             /           ______                                            /                                                   /             /           ______                                        /  361   I*\/ 9019      |            /             /           ______                                          /                                                   /             /           ______                                        /  361   I*\/ 9019   |  |            /             /           ______                                          /                                                   /             /           ______                                        /  361   I*\/ 9019   | 
            /   46        /  361   I*\/ 9019                 65                       /                                                   /   46        /  361   I*\/ 9019                 65                18*3 /   --- + ----------                                                                                                                                                                                                                                                                /   46        /  361   I*\/ 9019                 65                         /                                                   /   46        /  361   I*\/ 9019                 65                18*3 /   --- + ----------     |           /   46        /  361   I*\/ 9019                 65                       /                                                   /   46        /  361   I*\/ 9019                 65                18*3 /   --- + ----------  |  |           /   46        /  361   I*\/ 9019                 65                       /                                                   /   46        /  361   I*\/ 9019                 65                18*3 /   --- + ----------  | 
           /    -- + 2*3 /   --- + ----------  + --------------------------          /                                          27*      /    -- + 2*3 /   --- + ----------  + --------------------------       \/    216       54                                                                                                                                                                                                                                                           5*      /    -- + 2*3 /   --- + ----------  + --------------------------            /                                          27*      /    -- + 2*3 /   --- + ----------  + --------------------------       \/    216       54         |          /    -- + 2*3 /   --- + ----------  + --------------------------          /                                          27*      /    -- + 2*3 /   --- + ----------  + --------------------------       \/    216       54      |  |          /    -- + 2*3 /   --- + ----------  + --------------------------          /                                          27*      /    -- + 2*3 /   --- + ----------  + --------------------------       \/    216       54      | 
          /     9      \/    216       54                __________________         /                                                   /     9      \/    216       54                __________________                                                                                                                                                                                                                                                                                           /     9      \/    216       54                __________________           /                                                   /     9      \/    216       54                __________________                                  |         /     9      \/    216       54                __________________         /                                                   /     9      \/    216       54                __________________                               |  |         /     9      \/    216       54                __________________         /                                                   /     9      \/    216       54                __________________                               | 
         /                                              /           ______         /                                                   /                                              /           ______                                                                                                                                                                                                                                                                                           /                                              /           ______           /                                                   /                                              /           ______                                   |        /                                              /           ______         /                                                   /                                              /           ______                                |  |        /                                              /           ______         /                                                   /                                              /           ______                                | 
        /                                              /  361   I*\/ 9019         /                                                   /                                              /  361   I*\/ 9019                                                                                                                                                                                                                                                                                           /                                              /  361   I*\/ 9019           /                                                   /                                              /  361   I*\/ 9019                                    |       /                                              /  361   I*\/ 9019         /                                                   /                                              /  361   I*\/ 9019                                 |  |       /                                              /  361   I*\/ 9019         /                                                   /                                              /  361   I*\/ 9019                                 | 
       /                                         18*3 /   --- + ----------       /                                                   /                                         18*3 /   --- + ----------                                                                                                                                                                                                                                                                                         /                                         18*3 /   --- + ----------         /                                                   /                                         18*3 /   --- + ----------                                   |      /                                         18*3 /   --- + ----------       /                                                   /                                         18*3 /   --- + ----------                                |  |      /                                         18*3 /   --- + ----------       /                                                   /                                         18*3 /   --- + ----------                                | 
 7   \/                                             \/    216       54         \/                                                  \/                                             \/    216       54                                        35                                                                                                                       4                                                                                                                         \/                                             \/    216       54           \/                                                  \/                                             \/    216       54                                       |7   \/                                             \/    216       54         \/                                                  \/                                             \/    216       54                                    |  |1   \/                                             \/    216       54         \/                                                  \/                                             \/    216       54                                    | 
(- + ----------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------, - -- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------- + |- + ----------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------| *|- + ----------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------|)
 3                                      2                                                                                                                  2                                                                                3                                                                                                   __________________________________________________________________________________________________________________________________________                                       2                                                                                                                    2                                                                                \3                                      2                                                                                                                  2                                                                            /  \3                                      2                                                                                                                  2                                                                            / 
                                                                                                                                                                                                                                                                                                                                               /             __________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
                                                                                                                                                                                                                                                                                                                                              /             /           ______                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                             /   92        /  361   I*\/ 9019                                        380                                                   65                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
                                                                                                                                                                                                                                                                                                                                            /    -- - 2*3 /   --- + ----------  - -------------------------------------------------------------------------- - --------------------------                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                                                 _____________________________________________________________             /     9      \/    216       54                     _____________________________________________________________           __________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                                                /             __________________                                          /                                                   /             __________________                                        /           ______                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                               /             /           ______                                          /                                                   /             /           ______                                        /  361   I*\/ 9019                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
                                                                                                                                                                                                                                                              /   46        /  361   I*\/ 9019                 65                       /                                                   /   46        /  361   I*\/ 9019                 65                18*3 /   --- + ----------                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                             /    -- + 2*3 /   --- + ----------  + --------------------------          /                                          27*      /    -- + 2*3 /   --- + ----------  + --------------------------       \/    216       54                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                            /     9      \/    216       54                __________________         /                                                   /     9      \/    216       54                __________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                           /                                              /           ______         /                                                   /                                              /           ______                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                          /                                              /  361   I*\/ 9019         /                                                   /                                              /  361   I*\/ 9019                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                                         /                                         18*3 /   --- + ----------       /                                                   /                                         18*3 /   --- + ----------                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                   5   \/                                             \/    216       54         \/                                                  \/                                             \/    216       54                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                 - - + ----------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                   3                                      2                                                                                                                  2

Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:

x_{1} = \frac{7}{3} - \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} + \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} + \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}

x_{2} = \frac{7}{3} + \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} - \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2} + \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2}

Puntos máximos de la función:

x_{2} = \frac{7}{3} - \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} - \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} + \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}

x_{2} = \frac{7}{3} + \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} - \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} - \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}

Decrece en los intervalos

\left(-\infty, \frac{7}{3} - \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} - \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} + \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}\right] \cap \left(-\infty, \frac{7}{3} + \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} - \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} - \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}\right] \cap \left[\frac{7}{3} - \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} + \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} + \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}, \infty\right) \cap \left[\frac{7}{3} + \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} - \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2} + \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2}, \infty\right)

Crece en los intervalos

\left(-\infty, \frac{7}{3} - \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} + \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} + \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}\right] \cap \left(-\infty, \frac{7}{3} + \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} - \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2} + \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2}\right] \cap \left[\frac{7}{3} - \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} - \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} + \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}, \infty\right) \cap \left[\frac{7}{3} + \frac{\sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}{2} - \frac{\sqrt{\frac{92}{9} - 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}} - \frac{380}{27 \sqrt{\frac{46}{9} + \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}} + 2 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}} - \frac{65}{18 \sqrt[3]{\frac{361}{216} + \frac{\sqrt{9019} i}{54}}}}}{2}, \infty\right)

Puntos de flexiones

Hallemos los puntos de flexiones, para eso hay que resolver la ecuación

\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0

(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:

\frac{d^{2}}{d x^{2}} f{\left(x \right)} =

segunda derivada

2 \left(3 x - 2 + \frac{4}{\left(x - 4\right)^{3}}\right) = 0

Resolvermos esta ecuación
Raíces de esta ecuación

x_{1} = - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + \frac{250}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}} + \frac{50}{9}}}{2} + \frac{19}{6}

x_{2} = - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + \frac{250}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}} + \frac{50}{9}}}{2} - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}}{2} + \frac{19}{6}

Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:

x_{1} = 4

\lim_{x \to 4^-}\left(2 \left(3 x - 2 + \frac{4}{\left(x - 4\right)^{3}}\right)\right) = -\infty

\lim_{x \to 4^+}\left(2 \left(3 x - 2 + \frac{4}{\left(x - 4\right)^{3}}\right)\right) = \infty

- los límites no son iguales, signo

x_{1} = 4

- es el punto de flexión

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos

\left[- \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + \frac{250}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}} + \frac{50}{9}}}{2} - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}}{2} + \frac{19}{6}, - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + \frac{250}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}} + \frac{50}{9}}}{2} + \frac{19}{6}\right]

Convexa en los intervalos

\left(-\infty, - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + \frac{250}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}} + \frac{50}{9}}}{2} - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}}{2} + \frac{19}{6}\right] \cup \left[- \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + \frac{250}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}} + \frac{50}{9}}}{2} + \frac{19}{6}, \infty\right)

Asíntotas verticales

Hay:

x_{1} = 4

Asíntotas horizontales

Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo

\lim_{x \to -\infty}\left(\left(x^{2} \left(x - 2\right) - 5 x\right) + \frac{4}{x - 4}\right) = -\infty

Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda

\lim_{x \to \infty}\left(\left(x^{2} \left(x - 2\right) - 5 x\right) + \frac{4}{x - 4}\right) = \infty

Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha

Asíntotas inclinadas

Se puede hallar la asíntota inclinada calculando el límite de la función (x - 2)*x^2 - 5*x + 4/(x - 4), dividida por x con x->+oo y x ->-oo

\lim_{x \to -\infty}\left(\frac{\left(x^{2} \left(x - 2\right) - 5 x\right) + \frac{4}{x - 4}}{x}\right) = \infty

Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda

\lim_{x \to \infty}\left(\frac{\left(x^{2} \left(x - 2\right) - 5 x\right) + \frac{4}{x - 4}}{x}\right) = \infty

Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha

Paridad e imparidad de la función

Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:

\left(x^{2} \left(x - 2\right) - 5 x\right) + \frac{4}{x - 4} = x^{2} \left(- x - 2\right) + 5 x + \frac{4}{- x - 4}

- No

\left(x^{2} \left(x - 2\right) - 5 x\right) + \frac{4}{x - 4} = - x^{2} \left(- x - 2\right) - 5 x - \frac{4}{- x - 4}

- No
es decir, función
no es
par ni impar

Gráfico

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Gráfico de la función y = y=(x-2)x^2-5x+4/(x-4)

Gráfico:

Puntos de intersección:

Definida a trozos:

Solución