Question

The function $f\left(x\right)={\cot }^{-1}\left(x\right)+x$ increases in the interval

• A. $(1,\infty )$
• B. $(-1,\infty )$
• C. $(-\infty ,\infty )$
• D. $(0,\infty )$

Answer 1

The correct option is C

$(-\infty ,\infty )$

Explanation for the correct option :

Given function $f\left(x\right)={\cot }^{-1}\left(x\right)+x$

On differentiating w.r.t. $x$, we get,

$$\begin{gathered} f\text{'}\left(x\right)=-\frac{1}{(1+x^2)}+1 \\ =\frac{-1+1+x^2}{1+x^2} \\ =\frac{x^2}{1+x^2} \end{gathered}$$

Since, $f\text{'}\left(x\right)=\frac{x^2}{1+x^2}\ge 0$ for all real value of $x$.

Thus, $f\left(x\right)$ is strictly increasing for all real value of $x$.

Hence, the correct option is $\left(\mathrm{C}\right)$.