Let $g(x)=(x^3-8{)}^4.$

$($Section $4\mathrm{.}1,$ problem $24)$

$($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.

$1$

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