Answer
Absolute maximum: $f(3)=125$
Absolute minimum: $f(0)=-64$.
Work Step by Step
Using closed interval method, as $f$ in continuous over the given range, let's first calculate the value of $f$ at critical numbers followed by values of $f$ at endpoints of the given range.
For critical numbers, $f'(t)=0$ or $f'(t)$ should not exist. Let's differentiate $f(t)=(t^2-4)^3$ with respect to $t$ keeping $[-2,3]$ as interval.
$$\frac{df}{dt}=3(t^2-4)^2(2t)=6t(t^2-4)^2$$ $\frac{df}{dt}$ exists on the given interval. Thus, $\frac{df}{dt}=0$, $$6t(t^2-4)^2=0$$ Thus, $t=0$ and $(t^2-4)^2=0\implies t=-2,2$. Thus, $t=-2,0,2$ are critical numbers.
Thus, $f(0)=-64, f(-2)=f(2)=0$. At endpoints,
$f(-2)=0$ & $f(3)=125$. Thus, absolute maximum $=f(3)=125$ and absolute minimum $=f(0)=-64$.
