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Two real numbers $$x$$ and $$y$$ are varying such that their sum is 2009. Find the values of $$x$$ and $$y$$ such that their product is maximum. (Both values are equal; give that value.)
Correct Answer: 1004.5
With $$y = 2009 - x$$, $$xy = x(2009 - x) = \left(\frac{2009}{2}\right)^2 - \left(x - \frac{2009}{2}\right)^2$$. This is greatest when $$x = \frac{2009}{2}$$. So the product is maximum when $$x = y = 1004.5$$.
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