Question

If $y=2x+{\cot }^{-1}x+\log (\sqrt{1+x^2}-x),$ then y

• A. decreases on $(-\infty ,\infty )$
• B. decreases on $(0,\infty )$
• C. neither decreases nor increases on $(0,0)$
• D. increases on $(-\infty ,\infty )$

Answer 1

The correct option is D increases on $(-\infty ,\infty )$

$y=2x+co{t}^{-1}x+\log (\sqrt{x^2+1}-x)$
$\frac{dy}{dx}=2-\frac{1}{1+x^2}+\frac{\frac{x}{\sqrt{x^2+1}}-1}{\sqrt{x^2+1}-x}=2-\frac{2}{1+x^2}$
$\Rightarrow \frac{dy}{dx}=\frac{2x^2}{1+x^2}>0$ for all $x\in R$
Hence $y$ increases in $x\in (-\infty ,\infty )$