a: \(\dfrac{f\left(x_1\right)-f\left(x_2\right)}{x_1-x_2}=\dfrac{-x_1^2+2x_1+1+x_2^2-2x_2-1}{x_1-x_2}\)
\(=\dfrac{-\left(x_1-x_2\right)\left(x_1+x_2\right)+2\left(x_1-x_2\right)}{x_1-x_2}\)
\(=-\left(x_1+x_2\right)+2\)
Vì \(x_1,x_2\in\left(1;+\infty\right)\)
nên \(x_1>1;x_2>1\)
\(\Leftrightarrow\left(x_1+x_2\right)>2\)
\(\Leftrightarrow-\left(x_1+x_2\right)< -2\)
\(\Leftrightarrow-\left(x_1+x_2\right)+2< 0\)
hay f(x) nghịch biến trên khoảng \(\left(1;+\infty\right)\)
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