Question

Tìm GTNN và GTLN của biểu thức:

a) A = \(\frac{4x+3}{x^2+1}\)

b) B = \(\frac{x^2+x+1}{x^2+1}\)

Answer 1

a/ \(A=\frac{4x+3}{x^2+1}\Leftrightarrow Ax^2+A=4x+3\Leftrightarrow Ax^2-4x+A-3=0\)

\(\Delta'=4-A\left(A-3\right)\ge0\Leftrightarrow-A^2+3A+4\ge0\)

\(\Rightarrow-1\le A\le4\) \(\Rightarrow\left\{{}\begin{matrix}A_{max}=4\\A_{min}=-1\end{matrix}\right.\)

b/ \(B=\frac{x^2+x+1}{x^2+1}\Leftrightarrow Bx^2+B=x^2+x+1\Leftrightarrow\left(B-1\right)x^2-x+B-1=0\)

\(\Delta=1-4\left(B-1\right)^2\ge0\Leftrightarrow-4B^2+8B-3\ge0\)

\(\Rightarrow\frac{1}{2}\le B\le\frac{3}{2}\) \(\Rightarrow\left\{{}\begin{matrix}B_{max}=\frac{3}{2}\\B_{min}=\frac{1}{2}\end{matrix}\right.\)