Ta có : \(A=\frac{2010x+2680}{x^2+1}\)

\(=\frac{-335x^2-335+335x^2+2010x+3015}{x^2+1}\)

\(=-335+\frac{335\left(x+3\right)^2}{x^2+1}\ge-335\)

Dấu : \("="\)xảy ra khi và chỉ khi :

\(\frac{335\left(x+3\right)^2}{x^2+1}=0\)

\(\Leftrightarrow335\left(x+3\right)^2=0\)

\(\Leftrightarrow\left(x+3\right)^2=0\)

\(\Leftrightarrow x+3=0\Leftrightarrow x=-3\)

Vậy GTNN của \(A\)là : \(-335\Leftrightarrow x=-3\)

\(\text{Ta có:}A=\frac{2010x+2680}{x^2+1}=\frac{-335x^2+335x^2-335+2010x+2680+335}{x^2+1}.\)

             \(=\frac{-335\left(x^2+1\right)+335\left(x^2+6x+9\right)}{x^2+1}=-335+\frac{335\left(x+3\right)^2}{x^2+1}\ge-335\)

\(\text{Vậy GTNN của A=-335. Dấu bằng xảy ra khi và chỉ khi }x+3=0\Leftrightarrow x=-3\)

\(A=\frac{2010x+2680}{x^2+1}\)

\(=\frac{-355x^2-355+355x^2+2010x+3015}{x^2+1}\)

\(=-355+\frac{355.\left(x+3\right)^2}{x^2+1}\ge-335\)

Vậy  GTNN của A là -335 khi x=-3

\(A=\frac{2010x+2680}{x^2+1}\)

\(\Leftrightarrow Ax^2-2010x+A-2680=0\)

\(\Delta=\left(-2010\right)^2-4A\left(A-2680\right)\)

\(=-4\left(A-3015\right)\left(A+335\right)\)

Có nghiệm khi \(\Delta\ge0\)

\(\Leftrightarrow\left(A-3015\right)\left(A+335\right)\le0\)

\(\Leftrightarrow\hept{\begin{cases}A\le3015\\A\ge-335\end{cases}}\)

Ta có:A=\(\frac{335x^2+2010x+3015-\left(335x^2+335\right)}{x^2+1}\)

           =   \(\frac{335\left(x^2+6x+9\right)}{x^2+1}-\frac{335\left(x^2+1\right)}{x^2+1}\)

           =\(\frac{335\left(x+3\right)^2}{x^2+1}-335\)

Ta có: (x+3)²>= 0 

=>335(x+3)²>=0

Mà x²+1>0 

=>\(\frac{335\left(x+3\right)^2}{x^2+1}\ge0\)

=>\(\frac{335\left(x+3\right)^2}{x^2+1}-335\ge-335\)

=>A>= -335

Dấu "=" xảy ra <=> (x+3)²=0

                        <=> x+3=0

                          <=> x=-3

Vậy ...