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Question

Consider the function f : R → R defined by f(x)={2-sin(1/x)}|x|x00x=0

Then f is


A

monotonic on (0,) only

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B

not monotonic on (-,0) and(0,)

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C

monotonic on (-,0) only

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D

monotonic on (-,0)(0,)

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Solution

The correct option is B

not monotonic on (-,0) and(0,)


Explanation for the correct option:

Step 1. Definition of modulus function :

|x|=-xifx<0xifx>00ifx=0

Step 2. The function can be written as :

f(x)=-{2-sin(1/x)}xx<00x=0{2-sin(1/x)}xx>0

Step 3. Differentiate it with respect to x :

f'(x)=-x-cos1x-1x2-2-sin1xx<0x-cos1x-1x2+2-sin1xx>0f'(x)=-1xcos1x+sin1x-2x<01xcos1x+sin1x+2x>0

Thus, f is not monotonic on (-,0) and(0,).

Hence, the correct option is B.


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