Question 397039
We are given {{{a + b = 54}}} and want to minimize {{{a^2 + b^2}}}. Two ways to do this:


Solution 1:
We could square {{{a + b = 54}}} to obtain {{{a^2 + 2ab + b^2 = 2916}}}. We want to minimize {{{a^2 + b^2}}} and this is obtained when we maximize the value of {{{2ab}}}. If you've ever solved problems about rectangles having fixed perimeters, and know that the maximum area occurs when the rectangle is a square (many ways to prove this) then we deduce {{{a = b = 27}}}, and {{{a^2 + b^2 = 2916 - 2(27^2) = 1458}}}.


Solution 2:
By the Cauchy-Schwarz inequality,


{{{(a^2 + b^2)(1 + 1) >= (a + b)^2}}}


{{{2(a^2 + b^2) >= (a + b)^2 = 2916}}}


{{{a^2 + b^2 >= 1458}}}


Thus the minimal value is 1458. This occurs when a = b = 27.