A) \[(1,\,\,\infty )\]
B) \[(-1,\,\,\infty )\]
C) \[(-\infty ,\,\,\infty )\]
D) \[(0,\,\,\infty )\]
Correct Answer: C
Solution :
Key Idea: 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\[(-\infty ,\,\,\infty )\].
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