In the answer space below, provide one of the two positive integers whose sum is 200 and whose product is a maximum.

Yay for Calculus!

First, use the equation:

xy=S

This is x times y is equal to sum, or S.

Then, use the equation x + y = 200.

Then, solve for x or y. I will solve for y.

Doing so gives me the equation y = 200 - x

Next, use this value for y in the equation xy = S.

Give: x (200-x) = S

Simplify: 200x - x^2 = S.

Take the derivative: 200 - 2x = S'

Set it equal to 0 for critical numbers: 200 - 2x = 0

Solve: - 2x = - 200

Solve: 2x = 200

Sove: x = 100, which is one of your integers. The other is also 100.