pg. 820, #27

Find the global minimum and maximum of on the region
D = {(x,y) : |x| <= 1, |y| <= 1}.

Solution:

There are several critical points, but only one of them is in the domain |x| <= 1, |y| <= 1, for

.
But if , which is outside the specified domain. By the second derivative test
the function has a minimum at (x, y) = (0, 0):

A question arises: Does f(x,y) have an absolute or relative minimum at (0,0)?

Well, f(0,0) = 4 and as long as we have , so f has
an absolute minimum at (0, 0). A little analysis shows that the maximum value of 7 occurs at (-1, 1)
and (1, 1). . So, for all
(x,y) in the given domain. Furthermore, so the absolute maximum is taken on at these
two points.