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.

Question

x y and p

Answer 1

Let the numbers be x and 9-x, then p=x²(9-x). dp/dx=-x²+2x(9-x)=-3x²+18x=0 at an extremum.

So 3x(6-x)=0 and x=0 or 6. We can eliminate 0 which would make p=0. So p=36(3)=108, and the numbers are 6 and 3.