Mark the correct alternative in the following:
Question:

Mark the correct alternative in the following:

$f(x)=2 x-\tan ^{-1} x-\log \left\{x+\sqrt{x^{2}+1}\right\}$ is monotonically increasing when

A. $x>0$

B. $x<0$

C. $x \in R$

D. $X \in R-\{0\}$

Solution:

Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$

Given:-

$f(x)=2 x-\tan ^{-1} x-\log \left\{x+\sqrt{x^{2}+1}\right\}$

$\frac{d f(x)}{d x}=2-\frac{1}{1+x^{2}}-\frac{1}{\sqrt{x^{2}+1}}=f^{\prime}(x)$

For increasing function $f^{\prime}(x)>0$

$\Rightarrow 2-\frac{1}{1+x^{2}}-\frac{1}{\sqrt{x^{2}+1}}>0$

$x \in R$