Mathematics - Application of Derivative Question with Solution | TestHub

Since, $f(x)={\tan }^{-1}(\sin x+\cos x)$
$\therefore {\mathrm{f}}^{\prime }(\mathrm{x})=\frac{1}{1+(\sin \mathrm{x}+\cos \mathrm{x}{)}^2}(\cos \mathrm{x}-\sin \mathrm{x})$
$=\frac{\sqrt{2}\cos \left(x+\frac{\pi }{4}\right)}{1+(\sin x+\cos x{)}^2}$
$f(x)$ is increasing if $f^{\prime }(x)>0\Rightarrow \cos \left(x+\frac{\pi }{4}\right)>0$
$\Rightarrow -\frac{\pi }{2}<x+\frac{\pi }{4}<\frac{\pi }{2}\Rightarrow -\frac{3\pi }{2}<x+\frac{\pi }{4}$