In this exercise, we prove that for all x, y 0: a + b 1 1.40

In this exercise, we prove that for all x, y ≥ 0:

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a + b 1 1.40 y B where a 1 and 1 are numbers such that a +B- = 1. To do this, we prove that the function f(x,y) = ax + _xy xy satisfies f(x, y) 0 for all x, y 0. (a) Show that the set of critical points of f(x, y) is the curve y = x-1 (Figure 28). Note that this curve can also be described as x = -1. What is the value of f(x, y) at points on this curve? (b) Verify that the Second Derivative Test fails. Show, however, that for fixed b > 0, the function g(x) f(x.b) is concave up with a critical point at x = -1. (c) Conclude that for all x > 0, f(x, b) f(b-, b) = 0. (bp-1, b) inc y=xa-l inc Critical points of z=f(x, y) x =