Solution to AM - GM Introductory Problem 2
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 into our original expression that we wished to maximize, we get that
, which is our answer.