Find two numbers whose sum is 100 and whose product is a max

Find two numbers whose sum is 100 and whose product is a maximum.

Solution

Objective function: P(x,y) = xy Constraint Function: x+y = 100 Use constraint to get objective function in terms of one variable: x = 100 - y therefore: P(y) = (100-y)(y) P(y) = 100y - y^2 Differentiate and set to zero to solve for maximum/minimum: dP/dy = 100 - 2y = 0 100 = 2y y = 50 Therefore, from before: x = 100 - y x = 50 maximum product is: P = 50(50) P = 2500 BOL