You should investigate some proof techniques. The most simple ones are "direct proof" and "proof by contradiction".
For the direct proof, try to find an inequality that you know for sure holds true, and that is somehow connected to your problem (has a squareroot/square, two variables and a 2) and then play around with the inequality to try and get what you need to proove. (HINT: [MATH]x^2 \geq 0, \forall x \in R[/MATH])
For the proof by contradiction, suppose the opposite: [MATH]x,y>0,\ \ \sqrt{xy} > \frac{x+y}{2}[/MATH] and try to reach a contradiction.