Let $f(x)=x^4-4x^3.$

$($Section $4\mathrm{.}1,$ problem $12)$

$($a$)$ Use the derivative to find all critical points.

$($b$)$ Use the first or second derivative test to identify each critical point as a local minimum, a local maximum, or neither.

$($c$)$ Check your answers by graphing and provide a labeled sketch of $f(x),$ identifying the local maximum and minimum points.