| Directions: |
| The following questions consist of two statements, one labelled as "Assertion [A] and the other labelled as Reason [R]". You are to examine these two statements carefully and decide if Assertion [A] and Reason [R] are individually true and if so, whether the Reason [R] is the correct explanation for the given Assertion [A]. Select your answer from following options. |
| Let \[f\left( x \right)=2+cos\text{ }x\]for all real x |
| Statement-1 : For each real 't', then exist a point C in \[\left[ t,t+\pi \right]\] such that\[f'\left( C \right)\text{ }=\text{ }0\]. |
| Statement-2 : \[f\left( t \right)=f\left( t+2\pi \right)\]for each real t |
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 that \[f\left( x \right)=2+cos\,x\] Clearly \[f\left( x \right)\] is continuous and differentiable every where Also \[f'(x)=-sinx\Rightarrow f'(x=0)\] \[\Rightarrow -\sin x=0\Rightarrow x=n\pi \] \[\therefore \] These exists \[C\in [t,t+\pi ]\] for \[t\in R\] such that \[f'(C)=0\] \[\therefore \] Statement-1 is true Also \[f\left( x \right)\]being periodic function of period \[2\pi \] \[\therefore \]Statement-2 is true, but Statement-2 is not a correct explanation of Statement-1.
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