Answer
Local maximum: $3.25$ at $x=2.16$
Local minimum: $-4.05$ at $x=-1.16$
$f$ is increasing on: $(-1.16,2.16)$
$f$ is decreasing on: $(-4,-1.16)$ and $(2.16,5 )$
Work Step by Step
(graphed with desmos.com).
A local maximum at x=c exists if there is an interval around c such that f(c) is the largest function value on that interval.
A local minimum at x=c exists if there is an interval around c such that f(c) is the lowest function value on that interval.