Mark the correct alternative in the following:

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 $x \in(a, b)$

$f(x)=\frac{x}{1+|x|}$

For $x>0$

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

For $x<0$

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

Both are increasing for $f^{\prime}(x)>0$