Mark the correct alternative in the following:
Let $f(x)=\tan ^{-1}(g(x))$, where $g(x)$ is monotonically increasing for $0 A. increasing on $\left(0, \frac{\pi}{2}\right)$ B. decreasing on $\left(0, \frac{\pi}{2}\right)$ C. increasing on $\left(0, \frac{\pi}{4}\right)$ and decreasing on $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$ D. none of these
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:- $f(x)=\tan ^{-1}(g(x))$
$\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=\frac{\mathrm{g}^{\prime}(\mathrm{x})}{1+(\mathrm{g}(\mathrm{x}))^{2}}=\mathrm{f}(\mathrm{x})$
For increasing function
$f^{\prime}(x)>0$
$x \in\left(0, \frac{\pi}{2}\right)$
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