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2 numbers whose sum is 10 and the sum of their squares is a minimum.
If x is the first number, express the minimum as a function of x and find the numbers.
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Answer. The numbers are 5 and 5.
Proof
Let x be one number, then the other is 10-x.
The sum of their squares is
= = = = .
One can see that this quadratic function has the minimum at x = 5, and this minimum is equal to 50.
Thus we proved that one number, which gives the maximum, is 5.
Then the other number is 5 also.