\(VT\ge\sqrt{\left(x+y+z\right)^2+\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2}\left(minkowski\right)\)
\(\ge\sqrt{\left(x+y+z\right)^2+\left(\frac{9}{x+y+z}\right)^2}\left(cauchi-schwarz\right)\)
\(=\sqrt{\left(x+y+z\right)^2+\frac{1}{\left(x+y+z\right)^2}+\frac{80}{\left(x+y+z\right)^2}}\)
\(\ge\sqrt{2\sqrt{\left(x+y+z\right)^2.\frac{1}{\left(x+y+z\right)^2}}+\frac{80}{1}}\left(AM-GM\right)=\sqrt{82}\left(đpcm\right)\)
Dấu \("="\) xảy ra khi \(x=y=z=\frac{1}{3}\)
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