Problem

(a) Find the maximum value of the expression x^{2}y - xy^2 when 0\le x \le 1 and 0 \le y \le 1.

(b) Find the maximum value of the expression x^{2}y + y^{2}z + z^{2}x - x^{2}z - z^{2}y - y^{2}x when 0\le x \le 1, 0 \le y \le 1, and 0 \le z \le 1.

Solution

(a) Let f(x,y) = x^{2}y - xy^2 and check f(x,tx) = x^{3}(t - t^2) = x^{3}(-(1/2 - t)^2 + 1/4, so that f(x,tx) \ le x^{3}/4 \le 1/4, for 0\le x \le 1. The maximum of 1/4 is attained for x = 1, t = 1/2 or x = 1, y = 1/2.

(b) Let f(x,y,z) = x^{2}y + y^{2}z + z^{2}x - x^{2}z - z^{2}y - y^{2}x . Then f(x,sx,tx) = x^{3}(s + s^{2}t + t^2 - t - t^{2}s - s^2).

After a rearrangement, f(x,sx,tx) = x^{3}(s-t+t^2-s^2+s^{2}t-st^2) = x^{3}(s-t)(1-s)(1-t).

Let a = 1-s and b = 1-t. Then f(x,sx,tx) = x^{3}(a - b)ab = x^{3}(a^{2}b - ab^2), which is exactly the situation of part (a). With thus conclude that f(x,y,z) \le 1/4, with the maximum attained for x = 1 and a=1, b=1/2. In terms of s and t, s=0 and t=1/2. In terms of the original variables, x = 1, y = 0, z=1/2.