Answer
$ a.\quad$ critical points at $x=-2, 1$ and $3$.
$b.\quad $increasing on $(-2,1)\cup(3, \infty)$, decreasing on $(-\infty, -2)\cup(1,3)$.
$ c.\quad$ a local maximum at $x=1$ and local minima at $x=-2$ and at $x=3$
Work Step by Step
$(a)$
$f'$ is defined everywhere.
$f'(x)=0$ for $x=1,-2,3\qquad $... critical points at $x=-2, 1$ and $3$.
$(b)$
$\left[\begin{array}{lllll}
& (-\infty,-2) & (-2,1) & (1,3) & (3,\infty)\\
\text{test point} & -3 & 0 & 2 & 4\\
\text{evaluate }f' & -24 & 6 & -4 & 18\\
\text{sign of }f' & - & + & - & +\\
\text{behavior of }f(x) & \searrow & \nearrow & \searrow & \nearrow
\end{array}\right]$
f is increasing on $(-2,1)\cup(3, \infty)$, decreasing on $(-\infty, -2)\cup(1,3)$.
$(c)$
From the table, we see that f has:
- a local maximum at $x=1$ and
- local minima at $x=-2$ and at $x=3$
