Question

Determine the intervals of monotonicity of the function $f\left(x\right)=2x+{\cot }^{-1}x-\log \{x+\sqrt{(1+x^2)}\}$ is increasing.

Answer 1

Given, $f\left(x\right)=2x+co{t}^{-1}x-\log (x+\sqrt{(}1+x^2))$
For function to be monotical,it should be either increasing or decreasing.
For function to be increasing,$f^{\prime }\left(x\right)>0$
$\Rightarrow f^{\prime }\left(x\right)=2-\frac{1}{(1+x^2)}\frac{1}{\sqrt{(}1+x^2)}>0$ for all x $ϵ$ R,except x=0.
$\therefore$ The given function is monotically increasing for all x $ϵ$ R-{0}