[Math] Counting total number of local maxima and minima of a function

functionsmaxima-minima

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.

Best Answer

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.

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