Mark the correct alternative in the following:

Formula:- 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:-

$f(x)=\cos ^{-1} x+x$

$\mathrm{d}\left(\frac{\mathrm{f}(\mathrm{x})}{\mathrm{dx}}\right)=\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{2}}=\mathrm{f}^{\prime}(\mathrm{x})$

Now

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

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

$x \in R$

$\Rightarrow X \in(-\infty, \infty)$