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

Options

A. `R-{0}`
B. `R-{1}`
C. `4-{-1}`
D. `R-{-1,1}`

Answer

D

Explanation

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.