Question

cho x,y là 2 số thực dương 

CMR:

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

Answer 1

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

Ta có:

\(VT\le\frac{x\sqrt{y}+y\sqrt{x}}{2\sqrt{xy}}-\frac{x+y}{2}\)

\(=\frac{\sqrt{x}+\sqrt{y}}{2}-\frac{x+y}{2}\)

Giờ ta chỉ cần chứng minh 

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

\(\Leftrightarrow2x+2y-2\sqrt{x}-2\sqrt{y}+1\ge0\)

\(\Leftrightarrow\left(2x-2\sqrt{x}+\frac{1}{2}\right)+\left(2y-2\sqrt{y}+\frac{1}{2}\right)\ge0\)

\(\Leftrightarrow\left(\sqrt{2x}-\frac{1}{\sqrt{2}}\right)^2+\left(\sqrt{2y}-\frac{1}{\sqrt{2}}\right)^2\ge0\)(đúng)

Dấu = xảy ra khi \(x=y=\frac{1}{4}\)