Example 4. Investigate the monotonicity and extrema of the function $f(x)=\sqrt[3]{x^{2}}(x-1)$
Solution. $f(x)$ is defined for all $x \in \mathbb{R}$. By the product rule formula, $f^{\prime}(x)=\frac{2}{3 \sqrt[3]{x}}(x-1)+\sqrt[3]{x^{2}}=\frac{5 x-2}{3 \sqrt[3]{x}} ; x=\frac{2}{5}$ and $x=0$ are the critical points of $f(x)$. On the upper axis of the diagram (Fig. 7.5), the distribution of the signs of $f^{\prime}(x)$ is indicated, and on the lower axis, the behavior of the function $f(x)$ according to Theorems 6 and 7.

Answer. In the intervals $(-\infty ; 0) \cup\left(\frac{2}{5} ;+\infty\right) f(x)$ is monotonically increasing $\left(f^{\prime}(x)>0\right)$, in the interval $\left(0 ; \frac{2}{5}\right)$ it is monotonically decreasing, $x=0$ is a point of maximum with $y_{\max }=0, x=\frac{2}{5}$ is a point of minimum with $y_{\min }=-0.6 \sqrt[3]{0.16}$

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-085.jpg?height=176&width=451&top_left_y=1731&top_left_x=127)

Fig. 7.5

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-085.jpg?height=216&width=441&top_left_y=1691&top_left_x=707)

Fig. 7.6