Question

Find two positive numbers whose sum is 100 and the sum of whose squares is a minimum.

Answer 1

Let the two required numbers be x and 100 - x, then
Sum of its squares is given by S = x^2 + (100 - x)^2
For the sum of the squares to be minimum, dS/dx = 0

dS/dx = 2x - 2(100 - x) = 0
2x - 200 + 2x = 0
4x = 200
x = 50.

The two numbers is 50 and 50.