Counting total number of local maxima and minima of a function

Question

Find the total number of local maxima and local minima for the function $$ f(x) = \begin{cases} (2+x)^{3} &\text{if}\, -3 \lt x \le -1 \\ (x)^\frac{2}{3} &\text{if}\, -1 \lt x \lt 2 \end{cases} $$

My attempt : I differentiated the function for the two different intervals and obtained the following: $$ f'(x) = \begin{cases} 3\cdot(2+x)^{2} &\text{if}\, -3 \lt x \le -1 \\ \frac{2}{3}\cdot (x)^\frac{-1}{3} &\text{if}\, -1 \lt x \lt 2 \end{cases} $$ How do I obtain the maxima and minima points from here.

Any help will be appreciated.

Answer 1

You need to study $f'$ :

What is the sign of $f'(x), x\in ]-3 , 2 [$ ?

A local maxima/minima $x_m$ appears when :

• $f'(x_m) = 0$ or $x_m$ is a remarquable point (here $x_m \in \{-3,-1,2\}$)
• $f'$ changes sign before and after $x_m$

If the second condition is not verified, $x_m$ is an inflection point.