Answer
Absolute minimum is $0$ at $\theta=0,$
absolute maximum is $27$ at $\theta=-27$
Work Step by Step
To find absolute extrema of a continuous function f on a closed interval:
1. Evaluate $f$ at all critical points and endpoints.
2. Take the largest and smallest of these values.
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$h$ is continuous on $[-27,8]$.
Critical points:
$h'(\theta)=0$
$3\displaystyle \cdot\frac{2}{3}\theta^{-1/3}=0$
$\theta^{-1/3}=0\qquad $... none
$h'(\theta)$ is undefined for
$\theta=0 , \quad h(0)=0.$
Endpoints:
$h(-27)=3[(-27)^{1/3}]^{2}=3(-3)^{2}=27,$
$h(8)=(8^{1/3})^{2}=2^{2}=4$
Absolute minimum is $0$ at $\theta=0,$
absolute maximum is $27$ at $\theta=-27$
