Áp dụng BĐT BSC:
\(\sqrt{\dfrac{xy}{x+y+2z}}+\sqrt{\dfrac{yz}{y+z+2x}}+\sqrt{\dfrac{zx}{z+x+2y}}\)
\(=\dfrac{2\sqrt{xy}}{\sqrt{\left(1+1+2\right)\left(x+y+2z\right)}}+\dfrac{2\sqrt{yz}}{\sqrt{\left(1+1+2\right)\left(y+z+2x\right)}}+\dfrac{2\sqrt{zx}}{\sqrt{\left(1+1+2\right)\left(z+x+2y\right)}}\)
\(\le\dfrac{2\sqrt{xy}}{\sqrt{x}+\sqrt{y}+2\sqrt{z}}+\dfrac{2\sqrt{yz}}{\sqrt{y}+\sqrt{z}+2\sqrt{x}}+\dfrac{2\sqrt{zx}}{\sqrt{z}+\sqrt{x}+2\sqrt{y}}\)
\(\le\dfrac{1}{4}\left(\dfrac{2\sqrt{xy}}{\sqrt{x}+\sqrt{z}}+\dfrac{2\sqrt{xy}}{\sqrt{y}+\sqrt{z}}\right)+\dfrac{1}{4}\left(\dfrac{2\sqrt{yz}}{\sqrt{y}+\sqrt{x}}+\dfrac{2\sqrt{yz}}{\sqrt{z}+\sqrt{x}}\right)+\dfrac{1}{4}\left(\dfrac{2\sqrt{zx}}{\sqrt{z}+\sqrt{y}}+\dfrac{2\sqrt{zx}}{\sqrt{x}+\sqrt{y}}\right)\)
\(=\dfrac{1}{2}\left(\dfrac{\sqrt{xy}}{\sqrt{x}+\sqrt{z}}+\dfrac{\sqrt{xy}}{\sqrt{y}+\sqrt{z}}+\dfrac{\sqrt{yz}}{\sqrt{y}+\sqrt{x}}+\dfrac{\sqrt{yz}}{\sqrt{z}+\sqrt{x}}+\dfrac{\sqrt{zx}}{\sqrt{z}+\sqrt{y}}+\dfrac{\sqrt{zx}}{\sqrt{x}+\sqrt{y}}\right)\)
\(=\dfrac{1}{2}\left(\dfrac{\sqrt{xy}+\sqrt{yz}}{\sqrt{x}+\sqrt{z}}+\dfrac{\sqrt{xy}+\sqrt{zx}}{\sqrt{y}+\sqrt{z}}+\dfrac{\sqrt{yz}+\sqrt{zx}}{\sqrt{y}+\sqrt{x}}\right)\)
\(=\dfrac{1}{2}\left(\sqrt{x}+\sqrt{y}+\sqrt{z}\right)=\dfrac{1}{2}\)
Đẳng thức xảy ra khi \(x=y=z=\dfrac{1}{9}\)