\(\left(lna;lnb;lnc\right)=\left(x;y;z\right)\Rightarrow e^{x+y+z}=e^{\frac{1}{3}}\)
\(\Rightarrow x+y+z=\frac{1}{3}\Rightarrow z=\frac{1}{3}-x-y\)
\(T=2xy+7yz+3zx=2xy+\left(3x+7y\right)\left(\frac{1}{3}-x-y\right)\)
\(T=x-3x^2+\frac{7}{3}y-8xy-7y^2\)
\(\Leftrightarrow7y^2+\left(8x-\frac{7}{3}\right)y+3x^2-x+T=0\)
\(\Delta=\left(8x-\frac{7}{3}\right)^2-28\left(3x^2-x+T\right)\ge0\)
\(\Leftrightarrow-20x^2-\frac{28}{3}x+\frac{49}{9}-28T\ge0\)
\(\Rightarrow28T\le-20\left(x+\frac{7}{30}\right)^2+\frac{98}{15}\le\frac{98}{15}\)
\(\Rightarrow T\le\frac{7}{30}\Rightarrow T_{max}=\frac{7}{30}\) khi \(\left\{{}\begin{matrix}x=-\frac{7}{30}\\y=\frac{3}{10}\\z=\frac{4}{15}\end{matrix}\right.\)
