| Assertion [A]: The intervals in which\[f(x)=log\,\,cosx,\,\,0\le x\,\,\pi \] is decreasing is \[\left( 0,\frac{\pi }{2} \right)\]. |
| Reason [R]: The function \[\tan \,x\] is +ve in 3rd quadrant. |
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 \[f\left( x \right)=\log \,\cos \,x\,\Rightarrow f'\left( x \right)=\frac{1}{\cos \,x}.\left( -\sin \,x \right)\] For decreasing function, \[f'(x)<0\Rightarrow -\tan x<0\] \[\Rightarrow \tan x>0\] \[\Rightarrow x\in \left( 0,\frac{\pi }{2} \right).\] \[\therefore \] Assertion [A] is true We know that tan x is + ve in 3rd quadrant \[\therefore \] Reason (R) is true But it is not correct explanation of A Hence option [B] is the correct answer.
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