Answer
$ a.\quad$ critical points at $x=-5,-1,7$
$ b.\quad$ increasing on $(-5,-1)\cup(7, \infty)$, decreasing on $(-\infty, -5)\cup(-1,7)$.
$ c.\quad$ a local maximum at $x=-1$ and
local minima at $x=-5$ and at $x=7$
Work Step by Step
$(a)$
$f'$ is defined everywhere.
$f'(x)=0$ for $ x=-5,-1,7\qquad$ ... critical points
$(b)$
$\left.\begin{array}{ccccc}
& (-\infty,-5) & (-5,-1) & (-1,7) & (7,\infty)\\
\text{test point} & -6 & -2 & 0 & 8\\
\text{evaluate }f' & -65 & 27 & -35 & 117\\
\text{sign of }f' & - & + & - & +\\
\text{behavior of }f(x) & \searrow & \nearrow & \searrow & \nearrow
\end{array}\right.$
f is increasing on $(-5,-1)\cup(7, \infty)$, decreasing on $(-\infty, -5)\cup(-1,7)$.
$(c)$
From the table, we see that f has:
a local maximum at $x=-1$ and
local minima at $x=-5$ and at $x=7$
