Example 2. Let $F=-2 x^{3}+3 x^{2}+12 x-1$, find the extremum of $F$.


Solve: Here $a=-2, b=3$.
Let $x=y-(b / 3a)=y+(1 / 2)$, then
$$
F=f(y)=-2 y^{3}+\frac{27}{2} y+\frac{11}{2} \text{. }
$$

Here $a=-2, m=-27 / 2$. Thus, by Theorem 2, we know
$F$ maximum $=f(\sqrt{m} / 3a)=f(3 / 2)=19$;
$F_{\text{minimum}}=f(-\sqrt{m / 3a})=f(-3 / 2)=-8$.