Question

tìm m để f(x) = x² -2(m-1) + m-2 \(\le\) 0 \(\forall x\in\left[0;1\right]\)

Answer 1

\(a=1>0;\) \(\Delta'=\left(m-1\right)^2-m+2=m^2-3m+3=\left(m-\frac{3}{2}\right)^2+\frac{3}{4}>0\) ;\(\forall m\)

Để BPT thỏa mãn với \(\forall x\in\left[0;1\right]\Leftrightarrow x_1\le0< 1\le x_2\)

Đặt \(f\left(x\right)=x^2-2\left(m-1\right)x+m-2\)

\(\Rightarrow\left\{{}\begin{matrix}f\left(0\right)\le0\\f\left(1\right)\le0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}m-2\le0\\1-m\le0\end{matrix}\right.\)

\(\Rightarrow1\le m\le2\)