Question

cho hso \(y=\dfrac{x^3}{3}-x^2+mx+m-1\). tìm tất cả các tham số m để y'≥0, \(\forall x\in\left(1,3\right)\)

 

Answer 1

\(y'=x^2-2x+m\)

\(y'\ge0\) ; \(\forall x\in\left(1;3\right)\Leftrightarrow x^2-2x+m\ge0\) ;\(\forall x\in\left(1;3\right)\)

\(\Leftrightarrow m\ge\max\limits_{\left(1;3\right)}\left(-x^2+2x\right)\)

Xét hàm \(f\left(x\right)=-x^2+2x\) trên \(\left(1;3\right)\)

\(-\dfrac{b}{2a}=1\) ; \(f\left(1\right)=1\) ; \(f\left(3\right)=-3\)

\(\Rightarrow m\ge1\)