jmc

Question

Let f(x) = x(50 - x) + x(2 - x)for 0 x 2. Let M be the maximum value of f(x), and let x = x_0 be the point where the maximum value is attained. Enter the ordered pair (x_0,M).

Accepted Answer

By Cauchy-Schwarz, [ x(50 - x) + (2 - x)x ]^2 [(x + (2 - x))((50 - x) + x)] = 100,so f(x) 10. Equality occurs when x2 - x = 50 - xx.Cross-multiplying, we get x^2 = (2 - x)(50 - x) = x^2 - 52x + 100, so x = 10052 = 2513. Thus, (x_0,M) = ( 2513, 10 ).