A) \[(-\infty ,-1)\cup [1,\infty )\]
B) \[(-1,1)\]
C) \[(-\infty ,\infty )\]
D) none of these
Correct Answer: A
Solution :
[a] : Given, \[f(x)=\frac{x}{1+{{x}^{2}}}\] \[\Rightarrow \]\[f'(x)=\frac{(1+{{x}^{2}})(1)-(x)(2x)}{{{(1+{{x}^{2}})}^{2}}}\] \[=\frac{1+{{x}^{2}}-2{{x}^{2}}}{{{(1+{{x}^{2}})}^{2}}}=\frac{1-{{x}^{2}}}{{{(1+{{x}^{2}})}^{2}}}\] For decreasing function \[f'(x)<0,\frac{1-{{x}^{2}}}{{{(1+{{x}^{2}})}^{2}}}<0\] \[\Rightarrow \]\[1-{{x}^{2}}<0\]and \[{{(1+{{x}^{2}})}^{2}}>0\] ...(i) So, function decreases in the interval \[(-\infty ,-1]\cup [1,\infty )\]
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