The total number of local maxima and local minima of the function $f(x)=\begin{cases} (2+x)^{3}, & -3<x\le -1 \\ x^{2/3}, & -1<x<2 \end{cases}$ is

$f(x) = \begin{cases} (2+x)^{3}, & -3 < x \le -1 \\ x^{2/3}, & -1 < x < 2 \end{cases}$ $\text{To find: total number of local maxima and local minima}$ $1.\ \text{derivative and its sign on each piece}$ $2.\ \text{check the junction } x = -1$ $3.\ \text{count local maxima and minima}$ $\text{Local extremum: } f' \text{ changes sign at } x_0, \text{ or } f' \text{ undefined at } x_0$ $g(x) = (2+x)^{3} \Rightarrow g'(x) = 3(2+x)^{2} \ge 0$ $g'(-2) = 0 \text{ but } g' > 0 \text{ on both sides} \Rightarrow \text{no extremum at } x = -2$ $h(x) = x^{2/3} \Rightarrow h'(x) = \dfrac{2}{3}x^{-1/3} = \dfrac{2}{3\,x^{1/3}}$ $h'(0) \text{ is undefined (denominator } 0) \Rightarrow \text{cusp at } x = 0$ $h(0) = 0, \ h > 0 \text{ nearby} \Rightarrow \text{local min at } x = 0$ $\text{At } x = -1:\ (2-1)^{3} = 1 = (-1)^{2/3} \Rightarrow \text{continuous}$ $f(-1) = 1 > f(x) \text{ for } x \text{ nearby} \Rightarrow \text{local max at } x = -1$ $\text{max at } x=-1,\ \text{min at } x=0 \Rightarrow \text{total} = 2$ $\boxed{\text{Answer: (3) } 2}$