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$$\frac{\frac{1}{x \left(x - 2\right) \left(x - 1\right)} \left(6 x - 4\right) + \frac{\left(- x \left(x - 1\right) - \left(x - 2\right) \left(2 x - 1\right)\right) \left(\left(3 x^{2} - 4 x\right) - 2\right)}{x^{2} \left(x - 2\right)^{2} \left(x - 1\right)^{2}}}{1 + \frac{\left(\left(3 x^{2} - 4 x\right) - 2\right)^{2}}{x^{2} \left(x - 2\right)^{2} \left(x - 1\right)^{2}}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} + \frac{2}{3} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2}$$
$$x_{2} = - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2} + \frac{2}{3}$$
Signos de extremos en los puntos:
/ 2 \
| / ____________________________________________________________________________________________________________________ \ |
| | / _____________ | |
| | / / ____ | |
| | / 32 / 11 \/ 57 8 88 | |
| | / -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- | |
| | / 9 \/ 27 27 _____________ __________________________________________________ __________________________________________________| |
| | / / ____ / _____________ / _____________ | |
| | / / 11 \/ 57 / / ____ / / ____ | |
| | / 9*3 / -- + ------ / 16 / 11 \/ 57 8 / 16 / 11 \/ 57 8 | |
| | / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- / -- + 2*3 / -- + ------ + -------------------- | |
| | / / 9 \/ 27 27 _____________ / 9 \/ 27 27 _____________ | |
____________________________________________________________________________________________________________________ | ____________________________________________________________________________________________________________________ __________________________________________________ | / / / ____ / / ____ | |
/ _____________ | / _____________ / _____________ | / / / 11 \/ 57 / / 11 \/ 57 | |
/ / ____ | / / ____ / / ____ | / / 9*3 / -- + ------ / 9*3 / -- + ------ | |
/ 32 / 11 \/ 57 8 88 | 14 / 32 / 11 \/ 57 8 88 / 16 / 11 \/ 57 8 |2 \/ \/ \/ 27 27 \/ \/ 27 27 | |
/ -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- | - -- - 2* / -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- + 2* / -- + 2*3 / -- + ------ + -------------------- + 3*|- + ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------| |
/ 9 \/ 27 27 _____________ __________________________________________________ __________________________________________________ | 3 / 9 \/ 27 27 _____________ __________________________________________________ / 9 \/ 27 27 _____________ \3 2 2 / |
/ / ____ / _____________ / _____________ | / / ____ / _____________ / / ____ |
/ / 11 \/ 57 / / ____ / / ____ | / / 11 \/ 57 / / ____ / / 11 \/ 57 |
/ 9*3 / -- + ------ / 16 / 11 \/ 57 8 / 16 / 11 \/ 57 8 | / 9*3 / -- + ------ / 16 / 11 \/ 57 8 / 9*3 / -- + ------ |
/ \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- / -- + 2*3 / -- + ------ + -------------------- | / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- \/ \/ 27 27 |
/ / 9 \/ 27 27 _____________ / 9 \/ 27 27 _____________ | / / 9 \/ 27 27 _____________ |
/ / / ____ / / ____ | / / / ____ |
/ / / 11 \/ 57 / / 11 \/ 57 | / / / 11 \/ 57 |
/ / 9*3 / -- + ------ / 9*3 / -- + ------ | / / 9*3 / -- + ------ |
2 \/ \/ \/ 27 27 \/ \/ 27 27 | \/ \/ \/ 27 27 |
(- + ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------, atan|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|)
3 2 2 |/ ____________________________________________________________________________________________________________________ \ / ____________________________________________________________________________________________________________________ \ / ____________________________________________________________________________________________________________________ \|
|| / _____________ | | / _____________ | | / _____________ ||
|| / / ____ | | / / ____ | | / / ____ ||
|| / 32 / 11 \/ 57 8 88 | | / 32 / 11 \/ 57 8 88 | | / 32 / 11 \/ 57 8 88 ||
|| / -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- | | / -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- | | / -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- ||
|| / 9 \/ 27 27 _____________ __________________________________________________ __________________________________________________| | / 9 \/ 27 27 _____________ __________________________________________________ __________________________________________________| | / 9 \/ 27 27 _____________ __________________________________________________ __________________________________________________||
|| / / ____ / _____________ / _____________ | | / / ____ / _____________ / _____________ | | / / ____ / _____________ / _____________ ||
|| / / 11 \/ 57 / / ____ / / ____ | | / / 11 \/ 57 / / ____ / / ____ | | / / 11 \/ 57 / / ____ / / ____ ||
|| / 9*3 / -- + ------ / 16 / 11 \/ 57 8 / 16 / 11 \/ 57 8 | | / 9*3 / -- + ------ / 16 / 11 \/ 57 8 / 16 / 11 \/ 57 8 | | / 9*3 / -- + ------ / 16 / 11 \/ 57 8 / 16 / 11 \/ 57 8 ||
|| / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- / -- + 2*3 / -- + ------ + -------------------- | | / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- / -- + 2*3 / -- + ------ + -------------------- | | / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- / -- + 2*3 / -- + ------ + -------------------- ||
|| / / 9 \/ 27 27 _____________ / 9 \/ 27 27 _____________ | | / / 9 \/ 27 27 _____________ / 9 \/ 27 27 _____________ | | / / 9 \/ 27 27 _____________ / 9 \/ 27 27 _____________ ||
|| / / / ____ / / ____ | | / / / ____ / / ____ | | / / / ____ / / ____ ||
|| / / / 11 \/ 57 / / 11 \/ 57 | | / / / 11 \/ 57 / / 11 \/ 57 | | / / / 11 \/ 57 / / 11 \/ 57 ||
|| / / 9*3 / -- + ------ / 9*3 / -- + ------ | | / / 9*3 / -- + ------ / 9*3 / -- + ------ | | / / 9*3 / -- + ------ / 9*3 / -- + ------ ||
|| 4 \/ \/ \/ 27 27 \/ \/ 27 27 | | 1 \/ \/ \/ 27 27 \/ \/ 27 27 | |2 \/ \/ \/ 27 27 \/ \/ 27 27 ||
||- - + ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------|*|- - + ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------|*|- + ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------||
\\ 3 2 2 / \ 3 2 2 / \3 2 2 //
/ 2 \
| / ____________________________________________________________________________________________________________________\ |
| | / _____________ | |
| | / / ____ | |
| | / 32 / 11 \/ 57 8 88 | |
| | / -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- | |
| | __________________________________________________ / 9 \/ 27 27 _____________ __________________________________________________ | |
| | / _____________ / / ____ / _____________ | |
| | / / ____ / / 11 \/ 57 / / ____ | |
| | / 16 / 11 \/ 57 8 / 9*3 / -- + ------ / 16 / 11 \/ 57 8 | |
| | / -- + 2*3 / -- + ------ + -------------------- / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- | |
| | / 9 \/ 27 27 _____________ / / 9 \/ 27 27 _____________ | |
____________________________________________________________________________________________________________________ | __________________________________________________ ____________________________________________________________________________________________________________________ | / / ____ / / / ____ | |
/ _____________ | / _____________ / _____________ | / / 11 \/ 57 / / / 11 \/ 57 | |
/ / ____ | / / ____ / / ____ | / 9*3 / -- + ------ / / 9*3 / -- + ------ | |
/ 32 / 11 \/ 57 8 88 | 14 / 16 / 11 \/ 57 8 / 32 / 11 \/ 57 8 88 |2 \/ \/ 27 27 \/ \/ \/ 27 27 | |
/ -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- | - -- + 2* / -- + 2*3 / -- + ------ + -------------------- + 2* / -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- + 3*|- - ------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------| |
__________________________________________________ / 9 \/ 27 27 _____________ __________________________________________________ | 3 / 9 \/ 27 27 _____________ / 9 \/ 27 27 _____________ __________________________________________________ \3 2 2 / |
/ _____________ / / ____ / _____________ | / / ____ / / ____ / _____________ |
/ / ____ / / 11 \/ 57 / / ____ | / / 11 \/ 57 / / 11 \/ 57 / / ____ |
/ 16 / 11 \/ 57 8 / 9*3 / -- + ------ / 16 / 11 \/ 57 8 | / 9*3 / -- + ------ / 9*3 / -- + ------ / 16 / 11 \/ 57 8 |
/ -- + 2*3 / -- + ------ + -------------------- / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- | \/ \/ 27 27 / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- |
/ 9 \/ 27 27 _____________ / / 9 \/ 27 27 _____________ | / / 9 \/ 27 27 _____________ |
/ / ____ / / / ____ | / / / ____ |
/ / 11 \/ 57 / / / 11 \/ 57 | / / / 11 \/ 57 |
/ 9*3 / -- + ------ / / 9*3 / -- + ------ | / / 9*3 / -- + ------ |
2 \/ \/ 27 27 \/ \/ \/ 27 27 | \/ \/ \/ 27 27 |
(- - ------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------, atan|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|)
3 2 2 |/ ____________________________________________________________________________________________________________________\ / ____________________________________________________________________________________________________________________\ / ____________________________________________________________________________________________________________________\|
|| / _____________ | | / _____________ | | / _____________ ||
|| / / ____ | | / / ____ | | / / ____ ||
|| / 32 / 11 \/ 57 8 88 | | / 32 / 11 \/ 57 8 88 | | / 32 / 11 \/ 57 8 88 ||
|| / -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- | | / -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- | | / -- - 2*3 / -- + ------ - -------------------- + --------------------------------------------------------------- ||
|| __________________________________________________ / 9 \/ 27 27 _____________ __________________________________________________ | | __________________________________________________ / 9 \/ 27 27 _____________ __________________________________________________ | | __________________________________________________ / 9 \/ 27 27 _____________ __________________________________________________ ||
|| / _____________ / / ____ / _____________ | | / _____________ / / ____ / _____________ | | / _____________ / / ____ / _____________ ||
|| / / ____ / / 11 \/ 57 / / ____ | | / / ____ / / 11 \/ 57 / / ____ | | / / ____ / / 11 \/ 57 / / ____ ||
|| / 16 / 11 \/ 57 8 / 9*3 / -- + ------ / 16 / 11 \/ 57 8 | | / 16 / 11 \/ 57 8 / 9*3 / -- + ------ / 16 / 11 \/ 57 8 | | / 16 / 11 \/ 57 8 / 9*3 / -- + ------ / 16 / 11 \/ 57 8 ||
|| / -- + 2*3 / -- + ------ + -------------------- / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- | | / -- + 2*3 / -- + ------ + -------------------- / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- | | / -- + 2*3 / -- + ------ + -------------------- / \/ 27 27 27* / -- + 2*3 / -- + ------ + -------------------- ||
|| / 9 \/ 27 27 _____________ / / 9 \/ 27 27 _____________ | | / 9 \/ 27 27 _____________ / / 9 \/ 27 27 _____________ | | / 9 \/ 27 27 _____________ / / 9 \/ 27 27 _____________ ||
|| / / ____ / / / ____ | | / / ____ / / / ____ | | / / ____ / / / ____ ||
|| / / 11 \/ 57 / / / 11 \/ 57 | | / / 11 \/ 57 / / / 11 \/ 57 | | / / 11 \/ 57 / / / 11 \/ 57 ||
|| / 9*3 / -- + ------ / / 9*3 / -- + ------ | | / 9*3 / -- + ------ / / 9*3 / -- + ------ | | / 9*3 / -- + ------ / / 9*3 / -- + ------ ||
|| 4 \/ \/ 27 27 \/ \/ \/ 27 27 | | 1 \/ \/ 27 27 \/ \/ \/ 27 27 | |2 \/ \/ 27 27 \/ \/ \/ 27 27 ||
||- - - ------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------|*|- - - ------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------|*|- - ------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------||
\\ 3 2 2 / \ 3 2 2 / \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:
$$x_{1} = - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2} + \frac{2}{3}$$
Puntos máximos de la función:
$$x_{1} = - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} + \frac{2}{3} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2}$$
Decrece en los intervalos
$$\left[- \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2} + \frac{2}{3}, - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} + \frac{2}{3} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2}\right]$$
Crece en los intervalos
$$\left(-\infty, - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2} + \frac{2}{3}\right] \cup \left[- \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} + \frac{2}{3} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2}, \infty\right)$$