Problem: Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum. Answer:
Let variables x and y represent two nonnegative numbers. The sum of the two numbers is given to be 9 = x + y , so that y = 9  x . The product of x and the y^2 is a maximum. We can find the maximum of the function by finding the derivative. Let P be the product, so: P = x.y^2 . To find the maximum (or minimum) we have to differentiate. However, before we differentiate the righthand side, we have to write it as a function of x only. Substitute for y getting P = x.y^2 = x ( 9x)^2 . Now differentiate this equation using the product rule and chain rule, getting P' = x (2) ( 9x)(1) + (1) ( 9x)2 = ( 9x) [ 2x + ( 9x) ] = ( 9x) [ 93x ] = ( 9x) (3)[ 3x ] = 0 Which means that either x=9 or x=3 . Case 1: x = 3 If x = 3, then y = 9  3 = 6, and then P = x.y^2 = 3 * 6^2 = 3 * 36 = 108. Case 2: x = 9 If x = 9, then y = 9  9 = 0, and then P = x.0 = 0. Answer: the product of x and y^2 = 108.
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