Question

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

Answer 1

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`