Mathematics - Application of Derivative Question with Solution | TestHub

$f$ has domain $R$ and $f$ is even. $f^{\prime }(x)=\frac{2{\tan }^{-1}x}{1+x^2}-\frac{2x}{{\left(1+x^2\right)}^{3/2}}$

$$\begin{array}{rlr}& =\frac{2}{1+x^2}g(x)\text{ where }& g(x)={\tan }^{-1}x-\frac{x}{\sqrt{1+x^2}}\\ & \text{ Let }x=\tan \theta \text{ where }\theta \in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)& \\ & g(x)=\theta -\sin \theta & \\ & f^{\prime }(x)>0\text{ if }g(x)>0& \\ & \text{ i.e., if }\theta \in \left(0,\frac{\pi }{2}\right);x>0.& \\ & f\text{ is even }\therefore f^{\prime }(x)<0\text{ if }x<0.& \end{array}$$