Question

Cho \(y=\dfrac{x^3}{3}-x^2+\left(m-1\right)x+3\). Tìm \(m\) để \(y'\ge0\forall x\in\left(-1;+\infty\right)\).

Answer 1

\(y'=x^2-2x+(m-1) \geq0 \forall x \in (-1,+\infty) \)

\(\Rightarrow x^2-2x\ge1-m\forall x\in\left(-1,\infty\right)\)

Đặt \(f\left(x\right)=x^2-2x\Rightarrow f'\left(x\right)=2x-2\)

\(\Rightarrow f'\left(x\right)=0\Rightarrow x=1\)