What is the smallest possible value of the sum of their squares if the sum of two positive numbers is 16?

If the sum of two positive integers, ##x## and ##y## is ##16## ##x + y = 16## ##y = (16 - x)##

The sum of their squares is ##x^2 + (16 - x)^2## ##=2x^2 -32x +256##

The minimum will occur when the derivative ##= 0## i.e. when ##4x - 32 = 0## That is the minimum occurs at ##(x,y) = (8,8)## and the minimum possible value of the sum of the squares is ##8^2 + 8^2## ##=128##