Find two real numbers whose sum is 10 and whose product is maximal?
Let x and (10-x) be the two numbers. We must find x such that the product,
P(x) = x(10-x) (1)
is maximal. Expanding this equation we have
P(x) = 10x-x^2 (2)
As the reader can see, this is just a parabola opening downward, (I call this a frowning parabola ...
This shows how to find two numbers with maximum product and given sum.