find two nonnegative numbers whose sum is 9 such that the pr

find two nonnegative numbers whose sum is 9 such that the product of one number with the square of the other is maximum.

Let the numbers are x and y

x + y = 9

x = 9 - y

P = x y² = (9-y)y² = 9y²-y³

P \' = 18y - 3y²

P \' =0

3y(6 -y) =0

y =0 , 6

When y=6, x = 9-6 =3

So the numbers are 3 and 6