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