Answer
$ a.\quad$ critical points at $x=-2$ and $x=0$
$ b.\quad$ f is increasing on $(-\infty, -2)\cup(0, \infty)$, decreasing on $(-2,0)$.
$ c.\quad$ local maximum at $x=-2$,
local minimum at $x=0$
Work Step by Step
$(a)$
$f'$ is not defined at $ x=0\qquad$ ... critical point
(x=0 has not been excluded, so we take that f is defined at x=0)
$f'(x)=0$ for $ x=-2\qquad$ ... critical point
Critical points at $x=-2$ and $x=0$
$(b)$
$\left[\begin{array}{llll}
& (-\infty,-2) & (-2,0) & (0,\infty)\\
\text{test point, }t & -8 & -1 & 1\\
\text{evaluate }f'(t) & 64 & -1 & 1
\end{array}\right]$
$f':\stackrel{\nearrow}{++} \stackrel{-2}{|} \stackrel{\searrow}{- - } \quad \stackrel{0}{)(}$ $\stackrel{\nearrow}{++}$
f is increasing on $(-\infty, -2)\cup(0, \infty)$, decreasing on $(-2,0)$.
$(c)$
From the table, we see that f has:
local maximum at $x=-2$,
local minimum at $x=0$
