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.
Let variables x and y represent two nonnegative numbers. The sum of the two numbers is given to be
9 = x + y ,
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) [ -2x + ( 9-x) ]
= ( 9-x) [ 9-3x ]
= ( 9-x) (3)[ 3-x ]
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.