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