Prove that if the numbers x,y are of one sign, then $\displaystyle |\frac{x+y}{2}-\sqrt{xy}|+|\frac{x+y}{2}+\sqrt{xy}|=|x|+|y|$
I am sorry as I am new to this forum I didnt know that I have to show my work as well.. moreover this is not my homework what I have done
$\displaystyle |\frac{x+y}{2}-\sqrt{xy}|+|\frac{x+y}{2}+\sqrt{xy}|=|x|+|y|$
Solving R.HS. i.e. $\displaystyle |\frac{x+y}{2}-\sqrt{xy}|= |\frac{x+y -\sqrt{xy}}{2}| = \frac{(\sqrt{x}-\sqrt{y})^2}{2}$
$\displaystyle |\frac{x+y}{2}+\sqrt{xy}| = |\frac{ (\sqrt{x}+\sqrt{y})^2}{2} |$
Please suggest further..