Sr Examen
Otras calculadoras
- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^3+3*x^2+2
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-sqrt(3*x+1)
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x*(-1-log(x))
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x+1/(x-1)^2
-
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)
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:
$$x_{1} = 4$$
$$x_{1} = 4$$
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
$$x_{1} = -0.209431189721467$$
$$x_{2} = -0.209431189721563$$
$$x_{3} = -1.3316546931043$$
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
$$x_{1} = -0.209431189721467$$
$$x_{2} = -0.209431189721563$$
$$x_{3} = -1.3316546931043$$
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:
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)$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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:
2
__________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________ / __________________________________________________________________________________________________________________________________________\ / __________________________________________________________________________________________________________________________________________\
/ __________________ / __________________ | / __________________ | | / __________________ |
/ / ______ / / ______ | / / ______ | | / / ______ |
/ 92 / 361 I*\/ 9019 65 380 / 92 / 361 I*\/ 9019 65 380 | / 92 / 361 I*\/ 9019 65 380 | | / 92 / 361 I*\/ 9019 65 380 |
/ -- - 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 / 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 / --- + ---------- / 46 / 361 I*\/ 9019 65 |
/ -- + 2*3 / --- + ---------- + -------------------------- / \/ 216 54 27* / -- + 2*3 / --- + ---------- + -------------------------- 5* / -- + 2*3 / --- + ---------- + -------------------------- / \/ 216 54 27* / -- + 2*3 / --- + ---------- + -------------------------- | / -- + 2*3 / --- + ---------- + -------------------------- / \/ 216 54 27* / -- + 2*3 / --- + ---------- + -------------------------- | | / -- + 2*3 / --- + ---------- + -------------------------- / \/ 216 54 27* / -- + 2*3 / --- + ---------- + -------------------------- |
/ 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 65 380
/ -- - 2*3 / --- + ---------- - -------------------------- + --------------------------------------------------------------------------
_____________________________________________________________ / 9 \/ 216 54 __________________ _____________________________________________________________
/ __________________ / / ______ / __________________
/ / ______ / / 361 I*\/ 9019 / / ______
/ 46 / 361 I*\/ 9019 65 / 18*3 / --- + ---------- / 46 / 361 I*\/ 9019 65
/ -- + 2*3 / --- + ---------- + -------------------------- / \/ 216 54 27* / -- + 2*3 / --- + ---------- + --------------------------
/ 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 65 380 / 92 / 361 I*\/ 9019 65 380 | / 92 / 361 I*\/ 9019 65 380 | | / 92 / 361 I*\/ 9019 65 380 |
/ -- - 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 / / ______ / / ______ |
/ 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 / --- + ---------- / 46 / 361 I*\/ 9019 65 / 46 / 361 I*\/ 9019 65 |
/ \/ 216 54 27* / -- + 2*3 / --- + ---------- + -------------------------- / -- + 2*3 / --- + ---------- + -------------------------- / \/ 216 54 27* / -- + 2*3 / --- + ---------- + -------------------------- 5* / -- + 2*3 / --- + ---------- + -------------------------- | / \/ 216 54 27* / -- + 2*3 / --- + ---------- + -------------------------- / -- + 2*3 / --- + ---------- + -------------------------- | | / \/ 216 54 27* / -- + 2*3 / --- + ---------- + -------------------------- / -- + 2*3 / --- + ---------- + -------------------------- |
/ / 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 65 380
/ -- - 2*3 / --- + ---------- - -------------------------- + --------------------------------------------------------------------------
/ 9 \/ 216 54 __________________ _____________________________________________________________ _____________________________________________________________
/ / ______ / __________________ / __________________
/ / 361 I*\/ 9019 / / ______ / / ______
/ 18*3 / --- + ---------- / 46 / 361 I*\/ 9019 65 / 46 / 361 I*\/ 9019 65
/ \/ 216 54 27* / -- + 2*3 / --- + ---------- + -------------------------- / -- + 2*3 / --- + ---------- + --------------------------
/ / 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 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
$$\left(-\infty, - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + \frac{250}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}} + \frac{50}{9}}}{2} - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}}{2} + \frac{19}{6}\right] \cup \left[- \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + \frac{250}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}}} + 2 \sqrt[3]{\frac{\sqrt{561}}{27} + \frac{25}{27}} + \frac{25}{9}}} + \frac{50}{9}}}{2} + \frac{19}{6}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$x_{1} = 4$$
$$x_{1} = 4$$
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
$$\lim_{x \to -\infty}\left(\left(x^{2} \left(x - 2\right) - 5 x\right) + \frac{4}{x - 4}\right) = -\infty$$
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
$$\lim_{x \to -\infty}\left(\frac{\left(x^{2} \left(x - 2\right) - 5 x\right) + \frac{4}{x - 4}}{x}\right) = \infty$$
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
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