Determine where the function $f(x)=(x^{2}-1)^{2/3}$ is increasing and where it is decreasing.

Solution $f$ is continuous for all numbers $x$, and $$f^\prime (x)=\frac{2}{3}({x^{2}-1})^{-1/3}({2x})=\frac{4x}{3({x^{2}-1})^{1/3}}$$

The Increasing/Decreasing Function Test states that $f$ is increasing on intervals where $f^\prime (x) >0$ and decreasing on intervals where $f^\prime (x) <0.$ We solve these inequalities by using the numbers $-1,$ $0,$ and $1$ to form four intervals. Then we determine the sign of $f^\prime$ in each interval, as shown in Table 2. We conclude that $f$ is increasing on the intervals $(-1,0)$ and $(1,\infty)$ and that $f$ is decreasing on the intervals $(-\infty ,-1)$ and $(0,1)$.