Answer
Absolute maximum: $f(-2)=33$
Absolute minimum: $f(2)=-31$.
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, $f'(x)=0$ or $f'(x)$ should not exist. Let's differentiate $f(x)=3x^4-4x^3-12x^2+1$ with respect to $x$ keeping $[-2,3]$ as interval.
$$\frac{df}{dx}=12x^3-12x^2-24x$$ $\frac{df}{dx}$ exists on the given interval. Thus, $\frac{df}{dx}=0$, $$12x^3-12x^2-24x=0\implies x^3-x^2-2x=0$$ $$x(x^2-x-2)=0$$ Thus, $x=0$ or $x^2-x-2=0\implies x=-1,2$. Therefore, three critical numbers $x=-1,0,2$.
Thus, $f(-1)=-4$, $f(0)=1$ & $f(2)=-31$. At endpoints,
$f(-2)=33$ & $f(3)=28$. Thus, absolute maximum $=f(-2)=33$ and absolute minimum $=f(2)=-31$.
