What is the minimum value of the sum if the product of two numbers is K?

Answer 1

The answer depends on what restrictions are placed on K and the two numbers, p and k/p.

If the factors can be negative, then there is no minimum. p can be a very large negative number, with K/p being very small.

If K is positive and the factors must be positive, then the smallest sum is 2*sqrt(k).

To see this, let a and b be the two numbers, with a*b=k. Then b=k/a and the sum of the two numbers is S = a+k/a.

To find the values of a which lead to extreme values of S, take the derivative of a+k/a and set to zero. The derivative is 1-k/(a^2) = 0. Solving for a gives a = +-sqrt(k).

a=sqrt(k) is a local minimum, (and the only minimum if we're considering only positive values).

This gets a lot more complicated if K or the factors must be rational, or worse still, integers. Such limitations have not been made clear in the question.