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The domain of the derivative of the func...

The domain of the derivative of the function: `f(x)={{:(,tan^(-1)x,|x| le1),(,(1)/(2)(|x|-1),|x|gt1):}`

A

`R-{0}`

B

`R-{1}`

C

`4-{-1}`

D

`R-{-1,1}`

Text Solution

Verified by Experts

The correct Answer is:
D
We have
`f(x)={{:(,(1)/(2)(-x-1),x lt -1),(,tan^(-1)x,-1lexle1),(,(1)/(2)(x-1),x gt1):}`
We observe that
`underset(x to -1^(-))lim f(x)=underset(x to -1^(-))lim (1)/(2)(-x-1)=0`
`underset(x to -1^(-))lim f(x)=underset(x to -1^(+))lim tan^(-1)x=tan^(-1)=-pi//4`
Clearly, `underset(x to -1^(-))lim f(x) ne underset(x to -1^(+))lim f(x)`
So, f(x) is not continuous at x=-1
Similarly, f(x) is not continuous at x=1
Consequenctly f(x) is not differentiable at `x=pm1`
At all other points f(x) is differentiable.
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