A không có GTNN nhé bạn, chỉ có GTLN thôi.
Đặt \(a=2006\)
\(A=\frac{x}{\left(x+a\right)^2}=\frac{x}{x^2+2ax+a^2}\)
\(\Leftrightarrow A\cdot\left(x^2+2ax+a^2\right)=x\)
\(\Leftrightarrow x^2\cdot A+x\cdot2a\cdot A+a^2\cdot A-x=0\)
\(\Leftrightarrow x^2\cdot A+x\cdot\left(2Aa-1\right)+Aa^2=0\)
\(\Delta=\left(2aA-1\right)^2-4\cdot A\cdot Aa^2\)
\(=\left(2aA-1\right)^2-\left(2Aa\right)^2\)
\(=\left(2Aa-1-2Aa\right)\left(2Aa-1+2Aa\right)\)
\(=-1\cdot\left(4Aa-1\right)\ge0\)
\(\Leftrightarrow4Aa-1\le0\)
\(\Leftrightarrow A\le\frac{1}{4a}=\frac{1}{8024}\)
Dấu "=" xảy ra \(\Leftrightarrow x=2006\)
