Discovering the max/min, if $f'(x)=0$ seems to give only the other?
One can reason that the maximum of this occurs when $4-y^2$ is greatest. By differentiating one finds
$$(4-y^2)'=-2y := 0$$ exactly when $y=0$. By pluggin this in one discovers that it cannot be the minimum (since there are clearly smaller values), so it must be the maximum.
However, how can I discover the minimum with the derivative?
(one can of course find it by considering that $4-y^2 \ge 0$ must hold. So the minimum of this is (in this case seen to occur when $4-y^2=0$.