Question

How do you find the two positive real numbers whose sum is 40 and whose product is a maximum?

Answer 1

You have to find first a function to represent the problem stated, and then find a maximum of that function

Explanation:

The problem states that we are looking for two numbers `x` and `y` such as `x+y=40`, that is

`y=40-x`

We would like to find where the product `x*y` is maximum, but from the above equation we can write:

`x*y=x*(40-x) = -x^2+40x`.

So we now have a one-variable function `f(x)=-x^2+40x`, and must find a positive value of `x` where the function `f` reaches a maximum.

To do that we calculate the derivative `f'(x)=-2x+40`, and we look for values of `x` where `f'(x)=-2x+40=0`. There is only one such value (critical point) with `x=20`.

Now the second derivative `f''(x)=-2` is negative everywhere, and therefore is negative at the critical point `x=20`. Hence, `x=20` is a maximum for `f`.

But we also know that `y=40-x`, so the value of `y` is also `20`.

The solution is then `x=20, y=20`