Find the two positive numbers that satisfy the given requirements:

"The sum is S and the product is a maximum."

We are given with two conditions and we label each positve number as x and y. The sum of the 2 numbers is x + y equal to S. The product is maximum.

The product is xy. To find the maximum

P = xy = x (S - x) = Sx - x2

dP = S - 2x = 0

hence x = S/2

y then is equal to S/2 also.