Find the absolute maximum and minimum values of f on the set d. f(x, y) = 4x + 6y − x2 − y2 + 7, d = {(x, y) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 5}

One thing you can do is compute the Hessian:



We have , and for all points , which means (by the second partial derivative test) that any critical point of is the site of a local maximum.

The critical points occur where . We have




so that (2, 3) is the only critical point of , and it happens to fall within the region . At this point, we have .

Meanwhile, on the boundary:

If , then , which has a maximum of 9 when , and a minimum of 0 when . Then and .

If , then , which has a maximum of 16 when , and a minimum of 5 when . Then (and we already know ).

If , then , which has a maximum of 11 when , and a minimum of 0 when or . Then .

If , then , which has a maximum of 16 when , and a minimum of 12 when or . Then .

So, over the region , has an absolute maximum of 20 at (2, 3), and an absolute minimum of 7 at (0, 0).