# Solution to AM - GM Introductory Problem 2

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### Problem

Find the maximum of for all positive .

### Solution

We can rewrite the given expression as . To maximize the whole expression, we must minimize . Since is positive, so is . This means AM - GM will hold for and .

By AM - GM, the arithmetic mean of and is at least their geometric mean, or . This means the sum of and is at least . We can prove that we can achieve this minimum for + by plugging in by solving for .

Plugging in for our original expression that we wished to maximize, we get that , which is our answer.