TITLE: If both $a,b>0$, then $a^ab^b \ge a^bb^a$
QUESTION [35 upvotes]: Prove that $a^a \ b^b \ge a^b \ b^a$, if both $a$ and $b$ are positive.

REPLY [3 votes]: $$ a^a b^b \overset{?}{\ge} a^b b^a \\
(a/b)^a \overset{?}{\ge} (a/b)^b $$
Now if $a \gt b$ then this is obviously true, because the parenthese evaluates to some number $x$ with $1 \lt x$ and such a number to a higher power (on the lhs) is greater than that of a smaller power (on the rhs).       
But if $b \gt a$ then this is again obviously true, because the parenthese evaluates to $0 \lt x \lt 1 $ and such a number to a higher power (on the rhs) is smaller than that of a smaller power (at the lhs).