AGM inequality for four variables Prove that for any $x,y,z,w \ge 0$
$$\sqrt[4]{xyzw} \le \frac{x+y+z+w}{4}$$
I tried using AGM and doing $xy \cdot zw = (\frac{xy + zw}{2})^2$ but couldn't get the proof 

$(\sqrt{x}-\sqrt{y})^2\ge0$ => $x+y-2\sqrt{x}\sqrt{y}\ge0$ => $\frac{x+y}{2}\ge\sqrt{xy}$ 
Using this, we get 
$x+y\ge2\sqrt{xy}$ and $z+w\ge2\sqrt{zw}$ 
Adding both, we get 
$x+y+z+w\ge2\sqrt{xy}+2\sqrt{zw}$ => $\frac{x+y+z+w}{2}\ge\sqrt{xy}+\sqrt{zw}$
Now using formula again on RHS, we get 
$\sqrt{xy}+\sqrt{zw}\ge2(xyzw)^{\frac{1}{4}}$ 
Combining last two equations, we get the required answer.
