Question

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

Answer 1

The answer is that both of the numbers have to be `150`.

Let `x` is one of the two number, the other one is `300-x`, so the function product is:

`y=x(300-x)rArry=300x-x^2`, abd now let's find the maximum of the function:

`y'=300-2x` that is positive in `(-oo,150)` and zero in `150`.

So, before 150 the function grows, and after `150` the function decreases.

So the number `150` is the local maximum of the function.