Question

Show that the function $f\left(x\right)=2x+{\cot }^{-1}x-\log \{x+\sqrt{(1+x^2)}\}$ is increasing $\forall$ R

Answer 1

Given, $f\left(x\right)=2x+{\cot }^{-1}x-\log \{x+\sqrt{(1+x^2)}\}$
$\therefore \frac{dy}{dx}=2-\frac{1}{1+x^2}-\frac{1}{\sqrt{(1+x^2)}}$

$=\frac{1+2x^2-\sqrt{1+x^2}}{(1+x^2)}$

Since, $1+2x^2=1+x^2+x^2>\sqrt{1+x^2}\forall x\in R$

Hence $\frac{dy}{dx}=+ive\forall x\in R$
Therefore $f\left(x\right)$ is an increasing function in $(-\infty ,\infty )$.