prove that this equality is always right for each positive x and y. prove that this inequality is hold for each positive x,y. 
$x\over\sqrt{y}$ + $y\over\sqrt{x}$  $\ge$ $\sqrt{x}$ + $\sqrt{y}$  
I want a detailed way of solving the question. 

Without loss of generality you can assume $0<x<y$. Then use the rearrangement inequality:
$$
\sqrt x + \sqrt y = \frac{x}{\sqrt x} + \frac{y}{\sqrt y} \le 
\frac{x}{\sqrt y} + \frac{y}{\sqrt x}
$$
