Solution to AM - GM Introductory Problem 2
Find the maximum of for all positive .
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 into our original expression that we wished to maximize, we get that , which is our answer.