\(VT=\left(xyz+1\right)\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)+\dfrac{x}{z}+\dfrac{z}{y}+\dfrac{y}{x}\)
\(=yz+xz+xy+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{x}{z}+\dfrac{z}{y}+\dfrac{y}{x}\)
\(=\left(yz+xz+xy\right)+\left(\dfrac{x^2}{xz}+\dfrac{z^2}{yz}+\dfrac{y^2}{xy}\right)+\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)
\(\ge\left(yz+xz+xy\right)+\dfrac{\left(x+y+z\right)^2}{\left(xz+yz+xy\right)}+\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)
(bđt Cauchy Shwarz dạng Engel)
\(\ge2\left(x+y+z\right)+\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)
(bđt AM - GM)
\(=\left(x+y+z\right)+\left(x+y+z+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)
\(\ge\left(x+y+z\right)+6\sqrt[6]{x\times y\times z\times\dfrac{1}{x}\times\dfrac{1}{y}\times\dfrac{1}{z}}\)
\(=x+y+z+6=VP\left(\text{đ}pcm\right)\)
