Let x and y be the two numbers. Let's say that either x = y or x < y. To keep things relatively simple, we'll make both of these numbers positive.
We are told that they add to 50, so
x+y = 50
which solves to
y = 50-x
Their product is
x*y = x*(y) = x*(50-x) = 50x - x^2 = -x^2 + 50x
So the function
f(x) = -x^2 + 50x
will give us the product f(x) for the input x. Keep in mind that if we know what x is, then y = 50-x is used to find y.
The function f(x) graphs out a parabola that opens downward. Check out the diagram below. This means the very peak point of the parabola is what we're after. This is the vertex. At the vertex, we'll have f(x) maxed out as much as possible. This occurs at the point (25,625) meaning x = 25 and y = 50-x = 50-25 = 25 both multiply to x*y = 25*25 = 625, which is the largest possible product.
In short, this provides a visual means to confirm you have the correct answers for part b.
