\(\dfrac{9}{x+y+z}\le\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\le1\Rightarrow x+y+z\ge9\)
\(3\sqrt[3]{\dfrac{1}{xyz}}\le1\Rightarrow xyz\ge27\)
\(Q=\Sigma\dfrac{1}{x\left(\sqrt{2}-1\right)+x+y+z}\le\Sigma\dfrac{1}{x\left(\sqrt{2}-1\right)+9}\le\dfrac{\sqrt{2}\left(\sqrt{2}-1\right)}{2}\)
\(\)thật vậy tương đương
\(\sqrt{2}\left(\sqrt{2}-1\right)\left[x\left(\sqrt{2}-1\right)+9\right]\left[y\left(\sqrt{2}-1\right)+9\right]\left[z\left(\sqrt{2}-1\right)+9\right]\ge2\left[\left(x\left(\sqrt{2}-1\right)+9\right)\left(z\left(\sqrt{2}-1\right)+9\right)+\left(x\left(\sqrt{2}-1\right)+9\right)\left(y\left(\sqrt{2}-1\right)+9\right)+\left(y\left(\sqrt{2}-1\right)\left(z\left(\sqrt{ }\right)\right)+9\left(\right)\right)\left(\right)2-1\left(\right)+9\right]\)
\(\Leftrightarrow xyz\sqrt{2}\left(\sqrt{2}-1\right)^4+9\sqrt{2}\left(\sqrt{2}-1\right)^3\left(xy+yz+xz\right)+81\sqrt{2}\left(\sqrt{2}-1\right)^2\left(x+y+z\right)+729\sqrt{2}\left(\sqrt{2}-1\right)\ge2\left(\sqrt{2}-1\right)^2\left(xy+yz+xz\right)+36\left(\sqrt{2}-1\right)\left(x+y+z\right)+486\)
\(\)\(\Leftrightarrow xyz\sqrt{2}\left(\sqrt{2}-1\right)^4+9\sqrt{2}\left(\sqrt{2}-1\right)^3\left(xy+yz+xz\right)-2\left(\sqrt{2}-1\right)^2\left(xy+Yz+xz\right)+81\sqrt{2}\left(\sqrt{2}-1\right)^2\left(x+y+z\right)-36\left(\sqrt{2}-1\right)\left(x+y+z\right)+729\sqrt{2}\left(\sqrt{2}-1\right)-486\ge0\left(1\right)\)
\(\left(1\right)\) là đúng do:
\(xyz\sqrt{2}\left(\sqrt{2}-1\right)^4\ge27\sqrt{2}\left(\sqrt{2}-1\right)^4\)
\(9\sqrt{2}\left(\sqrt{2}-1\right)^3\left(xy+yz+xz\right)-2\left(\sqrt{2}-1\right)^2\left(xy+yz+xz\right)=\left(xy+yz+xz\right)\left[9\sqrt{2}\left(\sqrt{2}-1\right)^3-2\left(\sqrt{2}-1\right)^2\right]\ge3\sqrt[3]{27^2}\left[9\sqrt{2}\left(\sqrt{2}-1\right)^3-2\left(\sqrt{2}-1\right)^2\right]\)
\([81\sqrt{2}\left(\sqrt{2}-1\right)^2-36\left(\sqrt{2}-1\right)]\left(x+y+z\right)\ge9\left[81\sqrt{2}\left(\sqrt{2}-1\right)^2-36\left(\sqrt{2}-1\right)\right]\)
=>vế trái của (1)
\(\ge27\sqrt{2}\left(\sqrt{2}-1\right)^4+3\sqrt[3]{27^2}\left[9\sqrt{2}\left(\sqrt{2}-1\right)^3-2\left(\sqrt{2}-1\right)^2\right]+9\left[81\sqrt{2}\left(\sqrt{2}-1\right)^2-36\left(\sqrt{2}-1\right)\right]+729\sqrt{2}\left(\sqrt{2}-1\right)-486=0\)
do đó \(Q\le\dfrac{\sqrt{2}\left(\sqrt{2}-1\right)}{2}\)
