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Assertion [A]: The function \[f(x)={{x}^{3}}-12x\] is strictly  increasing in\[\left( -\infty ,-2 \right)\cup \left( 2,\text{ }\infty  \right)\].

Reason [R]: For strictly increasing function \[f'\left( x \right)>0\].

A) Both A and R are individually true and R is the correct explanation of A.

B) Both A and R are individually true and R is not the correct explanation of A.

C) 'A' is true but 'R' is false

D) 'A' is false but 'R' is true

E) Both A and R are false.

Correct Answer: A

Solution :

Given \[f\left( x \right)={{x}^{3}}-12x\] \[f(x)=3{{x}^{2}}-12=3\left( {{x}^{2}}-4 \right)=3(x-2)\left( x+2 \right)\] \[f'(x)=0\Rightarrow x=2,-2\] For strictly increasing function \[f'(x)>0\], \[\Rightarrow x\in \left( -\infty ,-2 \right)\cup \left( 2,\text{ }\infty  \right)~\] \[\Rightarrow \]Assertion A is true Also Reason (R) is true (Definition of strictly increasing function) Clearly R is correct explanation of A Hence option [A] is the correct answer.