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##

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