Finding two real numbers?

Question

Find two real numbers whose difference is 50 and whose product is a minimum. Enter the exact value of the smallest of the two numbers as your answer.

Answer 1

The two numbers are x and x+50. You want to minimize the quantity x(x+50). Setting the derivative equal to zero and solving, we have:

d(x(x+50))/dx = 0
x+50 + x = 0
2x = -50
x = -25

So setting x=-25 will minimize the product, and we are done.

Answer 2

This might be too obvious, but what about 0 and 50?