Question

Cho x,y là 2 số thực dương. CMR:

\(\dfrac{x\sqrt{y}+y\sqrt{x}}{x+y}-\dfrac{x+y}{2}\le\dfrac{1}{4}\)

Answer 1

\(VT=\dfrac{x\sqrt{y}+y\sqrt{x}}{x+y}-\dfrac{x+y}{2}\le\dfrac{\sqrt{2xy\left(x+y\right)}}{x+y}-\dfrac{x+y}{2}\)

\(\le\dfrac{\left(x+y\right)\sqrt{\dfrac{x+y}{2}}}{x+y}-\dfrac{x+y}{2}\). Cần cm \(\sqrt{\dfrac{x+y}{2}}-\dfrac{x+y}{2}\le\dfrac{1}{4}\)

Đặt \(x+y=t>0\) thì:

\(\sqrt{\dfrac{t}{2}}-\dfrac{t}{2}\le\dfrac{1}{4}\Leftrightarrow-\dfrac{1}{4}\left(\sqrt{2t}-1\right)^2\le0\) *Đúng*