Question

Tìm GTNN, GTLN của biểu thức:

A=x²+x+1/x²+2x+2

Answer 1

\(A=\frac{x^2+x+1}{x^2+2x+2}\Leftrightarrow A\left(x^2+2x+2\right)=x^2+x+1\)

\(\Leftrightarrow Ax^2+2Ax+2A-x^2-x-1=0\)

\(\Leftrightarrow x^2\left(A-1\right)+x\left(2A-1\right)+\left(2A-1\right)=0\)

\(\Delta=\left(2A-1\right)^2-4\left(2A-1\right)\left(A-1\right)\ge0\)

\(\Rightarrow\left(2A-1\right)^2-\left(4A-4\right)\left(2A-1\right)\ge0\)

\(\Rightarrow\left(2A-1\right)\left(2A-1-4A+4\right)\ge0\)

\(\Rightarrow\left(2A-1\right)\left(3-2A\right)\ge0\Rightarrow\frac{1}{2}\le A\le\frac{3}{2}\)