a.
\(f'\left(x\right)=3x^2+2ax+b=0\) có 2 nghiệm \(x=-1;x=1\)
\(\Rightarrow\left\{{}\begin{matrix}3+2a+b=0\\3-2a+b=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=0\\b=-3\end{matrix}\right.\)
\(\Rightarrow f\left(x\right)=x^3-3x\)
\(f\left(-1\right)=2\) ; \(f\left(1\right)=-2\) ; \(f\left(6\right)=198\)
\(\Rightarrow\max\limits_{\left[-1;6\right]}f\left(x\right)=198\) ; \(\min\limits_{\left[-1;6\right]}f\left(x\right)=-2\)
b.
\(g'\left(x\right)=\left(2x-2\right).f'\left(x^2-2x-3\right)=0\Rightarrow\left[{}\begin{matrix}2x-2=0\\f'\left(x^2-2x-3\right)=0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=1\\x^2-2x-3=-4\\x^2-2x-3=-3\end{matrix}\right.\)
\(\Rightarrow x=\left\{0;1;2\right\}\)
\(\Rightarrow g\left(x\right)\) đồng biến trên các khoảng \(\left(0;1\right)\) và \(\left(2;+\infty\right)\)
\(\Rightarrow\left[{}\begin{matrix}\left(m;m+1\right)\subset\left(0;1\right)\\\left(m;m+1\right)\subset\left(2;+\infty\right)\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}m=0\\m\ge2\end{matrix}\right.\)
