| Assertion [A]: The point of local maxima of function \[f\left( x \right)={{x}^{3}}-3x+3\text{ }\] is \[x=-1\]. |
| Reason [R]: For local minima, \[f''\left( x \right)\text{ }>\text{ }0.\] |
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: B
Solution :
Given \[f\left( x \right)={{x}^{3}}-3x+3\] \[f'\left( x \right)=3{{x}^{2}}-3\] For critical points, \[f'\left( x \right)=0\Rightarrow 3{{x}^{2}}-3=0\] \[\Rightarrow ~{{x}^{2}}=1\text{ }\Rightarrow x=\pm 1\Rightarrow x=1,-1\] Also \[~f''\left( x \right)=6x\] Clearly at \[x=-1,\text{ }f''\left( x \right)=-6<0\] \[\Rightarrow \,\,f\left( x \right)\]has local maxima at \[x=-1\] \[\Rightarrow \]Assertion A is true We know that for local minima, \[f''\left( x \right)\text{ }>\text{ }0\] \[\Rightarrow \] Reason R is true But it is not correct explanation of A Hence option [B] is the correct answer.
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