Question 1013376
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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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<U>Answer</U>. The numbers are 5 and 5.


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<U>Proof</U>


Let x be one number, then the other is 10-x.


The sum of their squares is 


{{{x^2 + (10-x)^2}}} = {{{x^2 + 100 - 20x + x^2}}} = {{{2x^2 - 20x + 100}}} = {{{2*(x^2 - 10x + 50)}}} = {{{2*(x-5)^2 + 50}}}.

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.
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