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 $x=1$.
$(b)$
$\left[\begin{array}{ccccccc}
& -\infty & & -2 & & 1 & & \infty\\
\text{test point} & & -3 & | & 0 & | & 2 & \\
\text{evaluate }f' & & (-4)^{2}(-1) & | & (-1)^{2}(2) & | & (2)^{2}(4) & \\
\text{sign of }f' & & - & | & + & | & + & \\
\text{behavior of }f(x) & & \searrow & min & \nearrow & & \nearrow &
\end{array}\right]$
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 $x=1$
