\(\dfrac{f\left(x_1\right)-f\left(x_2\right)}{x_1-x_2}\)
\(=\left(\dfrac{3x_1+1}{x_1-2}-\dfrac{3x_2+1}{x_2-2}\right):\left(x_1-x_2\right)\)
\(=\dfrac{3x_1x_2-6x_1+x_2-2-3x_1x_2+6x_2-x_1+2}{\left(x_1-2\right)\left(x_2-2\right)}:\left(x_1-x_2\right)\)
\(=\dfrac{-6\left(x_1-x_2\right)-\left(x_1-x_2\right)}{\left(x_1-2\right)\left(x_2-2\right)}\cdot\dfrac{1}{x_1-x_2}\)
\(=\dfrac{-7}{\left(x_1-2\right)\left(x_2-2\right)}\)
Vì \(x_1;x_2\in\left(2;+\infty\right)\)
nên \(\left\{{}\begin{matrix}x_1>2\\x_2>2\end{matrix}\right.\Leftrightarrow\left(x_1-2\right)\left(x_2-2\right)>0\)
\(\Leftrightarrow\dfrac{-7}{\left(x_1-2\right)\left(x_2-2\right)}< 0\)
=> Hàm số nghịch biến
