Question

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

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

Answer 1

Answer: (C)

A function is said to be increasing, if
$f^{\prime }(x)>0$ and it said be decreasing if $f^{\prime }(x)<0$ .
Given, $f(x)={\cot }^{-1}x+x$
$\Rightarrow$ $f^{\prime }(x)=-\frac{1}{1+x^2}+1$
$=\frac{x^2}{1+x^2}f(x)$ clearly, $f^{\prime }(x)>0$ for all $x$ .
So, $f(x)$ increases in $(-\infty ,\infty )$ .