If-x-y-gt-0-then-xy-x-y-2-x-y-2-xy-

Question Number 99645 by bramlex last updated on 22/Jun/20

I f x, y > 0 t h e n ∣ \sqrt{x y} - \frac{x + y}{2} ∣ + ∣ \frac{x + y}{2} + \sqrt{x y} ∣ =

Commented by bemath last updated on 22/Jun/20

since \sqrt{xy} > 0 & \frac{x + y}{2} > 0

so \frac{x + y}{2} + \sqrt{xy} > 0

case (1) if \sqrt{xy} - \frac{x + y}{2} > 0

then \sqrt{xy} - \frac{x + y}{2} + \frac{x + y}{2} + \sqrt{xy} = 2 \sqrt{xy}

case (2) if \sqrt{xy} - \frac{x + y}{2} < 0

then - \sqrt{xy} + \frac{x + y}{2} + \frac{x + y}{2} + \sqrt{xy} = x + y

Answered by Farruxjano last updated on 22/Jun/20

W h e n x, y > 0 \Rightarrow \frac{x + y}{2} + \sqrt{x y} > 0 \Rightarrow

∣ \frac{x + y}{2} + \sqrt{x y} ∣= \frac{x + y}{2} + \sqrt{x y}

W e k n o w i n e q u a l i t y K o s h i :

\frac{x + y}{2} ⩾ \sqrt{x y} w h e n x, y > 0 S o,

∣ \sqrt{x y} - \frac{x + y}{2} ∣= \frac{x + y}{2} - \sqrt{x y}

∣ \sqrt{x y} - \frac{x + y}{2} ∣ + ∣ \frac{x + y}{2} + \sqrt{x y} ∣ = \frac{x + y}{2} - \sqrt{x y} +

+ \frac{x + y}{2} + \sqrt{x y} = x + y ▴ .