Examples from Global extrema

Extrema on a rectangle

Find the maximum and minimum of

on

For interior candidates,

The equations $f_x=f_y=0$ give either $x=0$, hence $y=0$, or $y=1$, hence $x^2=6$, which is outside the rectangle. The only interior candidate is $(0,0)$, with

On the vertical edges $x=\pm1$,

The derivative is $-1+6y$, so the edge critical point is $y=1/6$, where

The edge endpoints give

On the horizontal edges,

The latter has minimum $3$ at $x=0$ and maximum $5$ at $x=\pm1$. Comparing all candidates gives

and

The minimum is interior, while the maximum occurs at corners.

Extrema on the closed unit disk

Let

on

The interior gradient is

Solving $\nabla f=\mathbf{0}$ gives $(0,0)$, with $f(0,0)=0$.

The boundary is the unit circle. Parameterise it by

Then

Since $-1\le\sin(2t)\le1$,

The maximum $3$ occurs at

and the minimum $-1$ occurs at

Comparing with the interior value gives

at the two same-sign boundary points and

at the two opposite-sign boundary points.