Question

The discriminant fxx fyy − f2xy is 0 at the origin for each of the following functions, so the second derivative test fails to determine the type of the critical point. Determine whether the function has a local maximum, minimum, or neither at the origin by imagining what the surface z = f (x, y) looks like. Briefly justify your reasoning in each case.

f (x, y) = x2y2

f (x, y) = 1 − x2y2

f (x, y) = xy2

f (x, y) = x3y3

Answer 1

(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.