Answer
Local maximum: $-1.91$ at $x=1.77$
Local minimum: $-18.89$ at $x=-3.77$
$f$ is increasing on: $(-3.77,1.77)$
$f$ is decreasing on: $(-6,-3.77)$ and $(1.77,4 )$
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.