Minimum locus of a squared function ($\underset{g=0}{\min }(g^2-c)=-c$)

If a real-valued function $g$ has a zero in the feasible region, then $g^2-c\ge -c$, with equality precisely on that zero set. The minimum can therefore occur along a curve or a higher-dimensional set rather than at isolated critical points. To maximize it on a compact feasible region, maximize $|g|$: both the largest and the smallest values of $g$ must be considered, since a large negative value also has a large square.