Calc find the maximum of f(x, y) = y2 + xy â x2 on the square 0 ⤠x ⤠6, 0 ⤠y ⤠6.

Presumably, . We have




so on this region, has only one critical point at (0, 0)[/tex]. Since this point lies on the boundary, we can ignore it for the moment.

Along the boundaries, we have four scenarios to consider:






In the first case, is increasing over , so we'll attain when a value of .

In the second case, attains a minimum when , and would be increasing to either side. This would mean at we get .

In the third case, attains a max at , which would yield .

In the fourth case, attains a max at , giving a value of .

This means the absolute maximum of over the square is attained along the boundary when , with a maximum value of .