The "AGM inequality" (arithmetic-geometric mean inequality) says that the geometric mean of a set of numbers is always less than or equal to the arithmetic mean or, as applied to two numbers, , and they are equal if and only if the numbers in the set are all the same.

To apply that to this situation, Let a= x and b= c-x. Then the geometric mean is and the arithmetic mean is . The AGM inequality says that . The right side is a constant, independent of x, so that is the "maximum" possible value of the left side and the left side takes that value when x= c-x.

Of course, will be maximum when is.

Here, by the way, is yet another way to prove that: . Since a square in never negative, that can never be larger than and will only be equal when x-c/2= 0.