Question

How do you find two positive numbers whose sum is 300 and whose product is a maximum?

Answer 1

Let `x` and `y` be the two numbers whose sum is 300.
`y = 300 -x`

and let `f(x)` be their product
`f(x) = x xx (300 - x)`
`f(x) = 300x - x^2`

The maximum occurs when
`(d f(x))/dx = 0`

`(d (fx))/dx = 300 - 2x`
The minimum occurs when
`300 - 2x = 0`
`x = 150`

(and, since `x + y = 300`,
`y = 150`)