**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 right-hand side, we have to write it as a function of

*x*only. Substitute for

*y*getting

*P*=

*x.y^*2

=

*x*( 9-

*x*)^2 .

Now differentiate this equation using the product rule and chain rule, getting

*P*' =

*x*(2) ( 9-

*x*)(-1) + (1) ( 9-

*x*)2

= ( 9-

*x*) [ -2

*x*+ ( 9-

*x*) ]

= ( 9-

*x*) [ 9-3

*x*]

= ( 9-

*x*) (3)[ 3-

*x*]

= 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.**