A) \[\left( -\infty ,\text{ }-1/2 \right)\cup \text{ }\left( 1/2,\text{ }\infty \right)\]
B) \[\left( -1/2,\,\,0 \right)\cup \left( 1/2,\text{ }\infty \right)\]
C) \[\left( 0,\,\,1/2 \right)\cup \left( 1/2,\text{ }\infty \right)\]
D) \[\left( -\infty ,\,-\,1/2 \right)\cup \left( 0,\text{ 1/2} \right)\]
Correct Answer: D
Solution :
\[f\left( x \right)\]will be increasing if \[f'\left( x \right)\ge 0\] \[\Rightarrow 12x\left( 4{{x}^{2}}-1 \right)>0\Rightarrow 12x\left( 2x-1 \right)\left( 2x+1 \right)>0\] \[\Rightarrow x>\frac{1}{2}\] or \[x<0\] and \[2x+1>0\] \[\Rightarrow x\in \left( \frac{1}{2},\infty \right)or\,\,\,x\,\,\in \left( -\frac{1}{2},0 \right)\] \[\Rightarrow x\in \left( -\frac{1}{2},0 \right)\cup \left( \frac{1}{2},\infty \right)\]
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