Find two positive numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum.

What are we trying to maximize? The product P of one number x and square of the other number y:

P=xy^2

With only this equation, the best way to maximize P is to let x and y be infinite. That choice of x and y does not meet all the requirements of the problem, however. We also have a constraint:

x+y=9

or

x = 9 - y

Substituting the constraint into the product equation:

P=(9-y)*y^2 = 9y^2- y^3

To maximize P, we will find the derivative and set it equal to zero:

dP/dy = 18y -3y^2= 0

18y - 3y^2=0

3y(6-y)=0

So, y = 0 or 6.

To determine the maximum, we need to check the value of the product P at both y = 0 and y = 6

If y = 0, then x = 9 (why?) and P = 0

If y = 6, then x = 3 and P = 108

So, the numbers that maximize xy^2 are x=3 and y=6.