\(y'=g\left(x\right)=3x^2-2\left(m+1\right)x-\left(2m^2-3m+2\right)\)
Để hàm số đồng biến trên khoảng đã cho
\(\Leftrightarrow g\left(x\right)\ge0;\forall x\ge2\)
\(\Delta'=\left(m+1\right)^2+3\left(2m^2-3m+2\right)=7\left(m-\frac{1}{2}\right)^2+\frac{21}{4}>0;\forall m\)
\(\Rightarrow\) Để \(g\left(x\right)\ge0;\forall x\ge2\Leftrightarrow x_1< x_2\le2\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(x_1-2\right)\left(x_2-2\right)\ge0\\\frac{x_1+x_2}{2}< 2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x_1x_2-2\left(x_1+x_2\right)+4\ge0\\x_1+x_2< 4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}-\frac{1}{3}\left(2m^2-3m+2\right)-\frac{4}{3}\left(m+1\right)+4\ge0\\\frac{2}{3}\left(m+1\right)< 4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}-2m^2-m+6\ge0\\2m< 10\end{matrix}\right.\) \(\Rightarrow-2\le m\le\frac{3}{2}\)
