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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x+27/x^3
-
(x^2-8)/(x-3)
-
x-2+4/(x-2)
-
x^2-9*x+14
-
Expresiones idénticas
- 19x- doce /((5x- cuatro)(x- trece))
- 19x menos 12 dividir por ((5x menos 4)(x menos 13))
- 19x menos doce dividir por ((5x menos cuatro)(x menos trece))
- 19x-12/5x-4x-13
- 19x-12 dividir por ((5x-4)(x-13))
-
Expresiones semejantes
- 19x-12/((5x+4)(x-13))
- 19x+12/((5x-4)(x-13))
- 19x-12/((5x-4)(x+13))
Gráfico de la función y = 19x-12/((5x-4)(x-13))
Solución
Ha introducido
[src]
12
f(x) = 19*x - ------------------
(5*x - 4)*(x - 13)
$$f{\left(x \right)} = 19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}$$
f = 19*x - 12*1/((x - 13)*(5*x - 4))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0.8$$
$$x_{2} = 13$$
$$x_{1} = 0.8$$
$$x_{2} = 13$$
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:
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{23}{5} + \frac{1327}{75 \sqrt[3]{\frac{174513}{2375} + \frac{2 \sqrt{638655033} i}{4275}}} + \sqrt[3]{\frac{174513}{2375} + \frac{2 \sqrt{638655033} i}{4275}}$$
o sea hay que resolver la ecuación:
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{23}{5} + \frac{1327}{75 \sqrt[3]{\frac{174513}{2375} + \frac{2 \sqrt{638655033} i}{4275}}} + \sqrt[3]{\frac{174513}{2375} + \frac{2 \sqrt{638655033} i}{4275}}$$
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
$$- \frac{12 \left(69 - 10 x\right)}{\left(x - 13\right)^{2} \left(5 x - 4\right)^{2}} + 19 = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
$$x_{2} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
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:
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
Puntos máximos de la función:
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
Decrece en los intervalos
$$\left(-\infty, - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}\right] \cup \left[- \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}, \infty\right)$$
Crece en los intervalos
$$\left[- \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}, - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{12 \left(69 - 10 x\right)}{\left(x - 13\right)^{2} \left(5 x - 4\right)^{2}} + 19 = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
$$x_{2} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
Signos de extremos en los puntos:
________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________________________________________________________________________________________________________
/ _____________________________________ / _____________________________________
/ / ________________ / / ________________
/ 7442 / 18598860004321 \/ 55796572722963 13845841 48 / 7442 / 18598860004321 \/ 55796572722963 13845841 48
/ ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- 19* / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + ---------------------------------------------------------------------------------------------------------------------
________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________
/ _____________________________________ / / ________________ / _____________________________________ / _____________________________________ / / ________________ / _____________________________________
/ / ________________ / / 18598860004321 \/ 55796572722963 / / ________________ / / ________________ / / 18598860004321 \/ 55796572722963 / / ________________
/ 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 / 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841
/ ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ 19* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------
/ 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________ / 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________
/ / ________________ / / / ________________ / / ________________ / / / ________________
/ / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963 / / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963
/ 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------ / 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------
69 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 1311 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 12
(-- - ------------------------------------------------------------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, ---- - --------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
10 2 2 10 2 2 / ________________________________________________________________________________________________________________________________________________________________________________________________________________________________\ / ________________________________________________________________________________________________________________________________________________________________________________________________________________________________\
| / _____________________________________ | | / _____________________________________ |
| / / ________________ | | / / ________________ |
| / 7442 / 18598860004321 \/ 55796572722963 13845841 48 | | / 7442 / 18598860004321 \/ 55796572722963 13845841 48 |
| / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- | | 5* / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- |
| ________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ | | ________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ |
| / _____________________________________ / / ________________ / _____________________________________ | | / _____________________________________ / / ________________ / _____________________________________ |
| / / ________________ / / 18598860004321 \/ 55796572722963 / / ________________ | | / / ________________ / / 18598860004321 \/ 55796572722963 / / ________________ |
| / 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 | | / 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 |
| / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ | | 5* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ |
| / 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________ | | / 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________ |
| / / ________________ / / / ________________ | | / / ________________ / / / ________________ |
| / / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963 | | / / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963 |
| / 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------ | | / 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------ |
| 61 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 | |61 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 |
|- -- - ------------------------------------------------------------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|-- - -------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\ 10 2 2 / \2 2 2 / ________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________________________________________________________________________________________________________
/ _____________________________________ / _____________________________________
/ / ________________ / / ________________
/ 7442 / 18598860004321 \/ 55796572722963 13845841 48 / 7442 / 18598860004321 \/ 55796572722963 13845841 48
/ ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- 19* / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + ---------------------------------------------------------------------------------------------------------------------
/ 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________
/ / ________________ / _____________________________________ / _____________________________________ / _____________________________________ / / ________________ / _____________________________________
/ / 18598860004321 \/ 55796572722963 / / ________________ / / ________________ / / ________________ / / 18598860004321 \/ 55796572722963 / / ________________
/ 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 / 3721 / 18598860004321 \/ 55796572722963 13845841 / 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841
/ \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ 19* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------
/ / 75 \/ 1218375000 676875 _____________________________________ / 75 \/ 1218375000 676875 _____________________________________ / 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________
/ / / ________________ / / ________________ / / ________________ / / / ________________
/ / / 18598860004321 \/ 55796572722963 / / 18598860004321 \/ 55796572722963 / / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963
/ / 11250*3 / -------------- + ------------------ / 11250*3 / -------------- + ------------------ / 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------
69 \/ \/ \/ 1218375000 676875 \/ \/ 1218375000 676875 1311 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 12
(-- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------, ---- - --------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
10 2 2 10 2 2 / ________________________________________________________________________________________________________________________________________________________________________________________________________________________________ \ / ________________________________________________________________________________________________________________________________________________________________________________________________________________________________\
| / _____________________________________ | | / _____________________________________ |
| / / ________________ | | / / ________________ |
| / 7442 / 18598860004321 \/ 55796572722963 13845841 48 | | / 7442 / 18598860004321 \/ 55796572722963 13845841 48 |
| / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- | | 5* / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- |
| / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________| | ________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ |
| / / ________________ / _____________________________________ / _____________________________________ | | / _____________________________________ / / ________________ / _____________________________________ |
| / / 18598860004321 \/ 55796572722963 / / ________________ / / ________________ | | / / ________________ / / 18598860004321 \/ 55796572722963 / / ________________ |
| / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 / 3721 / 18598860004321 \/ 55796572722963 13845841 | | / 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 |
| / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ | | 5* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ |
| / / 75 \/ 1218375000 676875 _____________________________________ / 75 \/ 1218375000 676875 _____________________________________ | | / 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________ |
| / / / ________________ / / ________________ | | / / ________________ / / / ________________ |
| / / / 18598860004321 \/ 55796572722963 / / 18598860004321 \/ 55796572722963 | | / / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963 |
| / / 11250*3 / -------------- + ------------------ / 11250*3 / -------------- + ------------------ | | / 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------ |
| 61 \/ \/ \/ 1218375000 676875 \/ \/ 1218375000 676875 | |61 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 |
|- -- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------|*|-- - -------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\ 10 2 2 / \2 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:
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
Puntos máximos de la función:
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
Decrece en los intervalos
$$\left(-\infty, - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}\right] \cup \left[- \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}, \infty\right)$$
Crece en los intervalos
$$\left[- \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}, - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}\right]$$
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
$$\frac{24 \left(5 - \frac{5 \left(10 x - 69\right)}{5 x - 4} - \frac{10 x - 69}{x - 13}\right)}{\left(x - 13\right)^{2} \left(5 x - 4\right)^{2}} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{24 \left(5 - \frac{5 \left(10 x - 69\right)}{5 x - 4} - \frac{10 x - 69}{x - 13}\right)}{\left(x - 13\right)^{2} \left(5 x - 4\right)^{2}} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
$$x_{1} = 0.8$$
$$x_{2} = 13$$
$$x_{1} = 0.8$$
$$x_{2} = 13$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
$$\lim_{x \to -\infty}\left(19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}\right) = \infty$$
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 19*x - 12*1/((x - 13)*(5*x - 4)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}}{x}\right) = 19$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = 19 x$$
$$\lim_{x \to \infty}\left(\frac{19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}}{x}\right) = 19$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = 19 x$$
$$\lim_{x \to -\infty}\left(\frac{19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}}{x}\right) = 19$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = 19 x$$
$$\lim_{x \to \infty}\left(\frac{19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}}{x}\right) = 19$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = 19 x$$
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:
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = - 19 x - \frac{12}{\left(- 5 x - 4\right) \left(- x - 13\right)}$$
- No
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = 19 x + \frac{12}{\left(- 5 x - 4\right) \left(- x - 13\right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = - 19 x - \frac{12}{\left(- 5 x - 4\right) \left(- x - 13\right)}$$
- No
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = 19 x + \frac{12}{\left(- 5 x - 4\right) \left(- x - 13\right)}$$
- No
es decir, función
no es
par ni impar