Answer
$ a.\quad$ critical points at $x=-2$ and $x=1$.
$ b.\quad$ increasing on $(-2,1)$ and $(1, \infty)$, decreasing on $(-\infty, -2)$
$ c.\quad$ local minimum at $x=-2$ and
no local maximum.
Work Step by Step
$(a)$
$f'$ is defined everywhere.
$f'(x)=0$ for $ x=1,-2\qquad$ ... critical points at $x=-2$ and $x=1$.
$(b)$
$\left.\begin{array}{ccccccccc}
& -\infty & & -2 & & 1 & & \infty\\
\text{test point} & & -3 & | & 0 & | & 2 & \\
\text{evaluate }f' & & (-4)^{2}(-1) & | & (-1)^{2}(2) & | & (2)^{2}(4) & \\
\text{sign of }f' & & - & | & + & | & + & \\
\text{behavior of }f(x) & & \searrow & min & \nearrow & & \nearrow &
\end{array}\right.$
f is increasing on $(-2,1)$ and $(1, \infty)$, decreasing on $(-\infty, -2)$
$(c)$
From the table, we see that f has:
a local maximum at $x=0$ and
a local minimum at $x=1$
