Among all pairs of numbers with a sum of 75, find the pair whose product is maximum. Write your answers as fractions reduced to lowest terms.

Answer 1

75/2 + 75/2 = 75

(75/2)^2 = 5625/4 = 1406.25 = maximum product of two numbers summing to 75

75/2 = 37 1/2

37x38 = 1406

36x39 = 1404

the more different the two numbers, the smaller their product. Maximum product is when they are the same number, 75/2 and 75/2

x+y = 75

y=75-x

xy = x(75-x) = 75x -x^2

x^2 - 75x = A

A' = 2x -75 = 0

2x = 75

x = 75/2 = 37 1/2