# Important information about Maximization

Find two real numbers whose sum is 10 and whose product is maximal?

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#### Solution Preview

Let x and (10-x) be the two numbers. We must find x such that the product,

P(x) = x(10-x) (1)

is maximal. Expanding this equation we have

P(x) = 10x-x^2 (2)

As the reader can see, this is just a parabola opening downward, (I call this a frowning parabola ...

#### Solution Summary

This shows how to find two numbers with maximum product and given sum.

$2.19