Mark the correct alternative in the following:

Question:

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

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:- $f(x)=\tan ^{-1}(g(x))$

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

For increasing function

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

$x \in\left(0, \frac{\pi}{2}\right)$

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