SOLUTION: Find two numbers whose sum is 54 such that the sum of their squares is a minimum. (If a solution has a multiplicity of two, enter it in consecutive answer boxes.) smaller number=

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Question 397039: Find two numbers whose sum is 54 such that the sum of their squares is a minimum. (If a solution has a multiplicity of two, enter it in consecutive answer boxes.)
smaller number=
larger number=
Answer by richard1234(7193) (Show Source):
You can put this solution on YOUR website!
We are given and want to minimize . Two ways to do this:

Solution 1:
We could square to obtain . We want to minimize and this is obtained when we maximize the value of . 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 , and .

Solution 2:
By the Cauchy-Schwarz inequality,

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