2 ) Ta có :
\(f\left(x\right)=x^2-\left(2m+3\right)x+m^2-1\ge\frac{2017}{4}\)
\(\Leftrightarrow x^2-\left(2m+3\right)x+m^2-\frac{2021}{4}\ge0\)
Hiển nhiên dấu bằng sẽ xảy ra
\(\Delta=\left(2m+3\right)^2-4\left(m^2-\frac{2021}{4}\right)=0\)
\(\Leftrightarrow4m^2+12m+9-4m^2+2021=0\)
\(\Leftrightarrow12m+2030=0\)
\(\Leftrightarrow m=-\frac{1015}{6}\)
Để pt có 2 nghiệm dương phân biệt:
\(\left\{{}\begin{matrix}\Delta>0\\S>0\\P>0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}\left(2m+3\right)^2-4\left(m^2-1\right)>0\\2m+3>0\\m^2-1>0\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}-\frac{13}{12}< m< -1\\m>1\end{matrix}\right.\)
\(f\left(x\right)=x^2-\left(2m+3\right)x+m^2-1\)
\(f\left(x\right)=x^2-2\left(m+\frac{3}{2}\right)x+\left(m+\frac{3}{2}\right)^2-3m-\frac{5}{4}\)
\(f\left(x\right)=\left(x-m-\frac{3}{2}\right)^2-3m-\frac{5}{4}\ge-3m-\frac{5}{4}\)
\(\Rightarrow-3m-\frac{5}{4}=\frac{2017}{4}\Rightarrow-3m=\frac{1011}{2}\Rightarrow m=-\frac{337}{2}\)
