\(y=\dfrac{1}{3}\left(m^2-2m\right)x^3+mx^2+3x\)
=>\(y'=\dfrac{1}{3}\cdot\left(m^2-2m\right)\cdot3x^2+m\cdot2x+3\)
=>\(y'=\left(m^2-2m\right)x^2+2m\cdot x+3\)
Để hàm số đồng biến trên R thì y'>=0 với mọi x
=>\(\left\{{}\begin{matrix}\text{Δ}< =0\\a>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(2m\right)^2-4\left(m^2-2m\right)\cdot3< =0\\m^2-2m>0\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}4m^2-12\left(m^2-2m\right)< =0\\m\left(m-2\right)>0\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}-8m^2+24m< =0\\m\left(m-2\right)>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}-8\left(m^2-3m\right)< =0\\m\left(m-2\right)>0\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}m\left(m-3\right)>=0\\m\left(m-2\right)>0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}m>=3\\m< 0\end{matrix}\right.\)
