Question

Let $f\left(x\right)=2x+{\cot }^{-1}x+\ln (\sqrt{1+x^2}-x)$ then $f\left(x\right)$

• A. Increases $\forall xϵR$
• B. Decreases $\forall xϵR$
• C. Increases in $(0,\infty )$ & decreases in $(-\infty ,0)$
• D. Decreases in $(0,\infty )$ & increases in $(-\infty ,0)$

Answer 1

The correct option is A Increases $\forall xϵR$

$y=2x+{\cot }^{-1}x+\log (\sqrt{x^2+1}-x)$
$\frac{dy}{dx}=2-\frac{1}{1+x^2}+\frac{\frac{x}{\sqrt{x^2+1}}-1}{\sqrt{x^2+1}-x}=2-\frac{2}{1+x^2}$
$\Rightarrow \frac{dy}{dx}=\frac{2x^2}{1+x^2}>0$ for all $xϵR$
Hence $y$ increases for all $xϵR$