Question 1070500: The sum of two numbers is 8. Find the numbers if the sum of their squares is a minimum. (hint completing the square) Answer by ikleyn(52747) (Show Source):
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The sum of two numbers is 8. Find the numbers if the sum of their squares is a minimum. (hint completing the square)
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Let x be one of the two numbers.
Then the other number is 8-x, and the sum of their squares is
q(x) = (1)
= = .
This quadratic function has the minimum at x = = = = 4.
Calculate this maximum value. For it, substitute the value x= 4 into the function (1):
q(4) = = 16 + 16 = 32.
Answer. The minimum is achieved at x = 4 and is equal to 16.
The vertex of the quadratic function is (4,16).
