(a) We have f (0, 0) = 0 and f (x, y) = x2y2≥0 for all x, y. Therefore (0, 0) is a minimum.
(b) We have f (0, 0) = 1 and
f (x, y) = 1 − x2 y2≤1
for all x, y. Therefore (0, 0) is a maximum.
(c) We have f (0, 0) = 0. If x > 0 and y≠0, then f (x, y) > 0, while for x < 0 and y≠ 0 we have f (x, y) < 0. Thus (0, 0) is a saddle point.
(d) We have f (0, 0) = 0. If x > 0 and y > 0, then f (x, y) > 0, while for x > 0 and y < 0 we have f (x, y) < 0. Thus (0, 0) is a saddle point.