Difference between revisions of "Solution to AM - GM Introductory Problem 2"
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Revision as of 21:14, 13 May 2020
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.
