TITLE: Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
QUESTION [8 upvotes]: OK guys I have this problem:
For $x,y,p,q>0$ and $ \frac {1} {p} + \frac {1}{q}=1 $ prove that $ xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
It says I should use Jensen's inequality, but I can't figure out how to apply it in this case. Any ideas about the solution?

REPLY [6 votes]: By AM-GM
$$\frac{x^p}{p}+\frac{y^q}{q}\geq\left(x^p\right)^{\frac{1}{p}}\left(y^q\right)^{\frac{1}{q}}=xy$$