The set of all points, where the function `f(x)=x/(1+|x|)` is differentiable, is given by

Options

A. `(-oo,oo)`
B. `(0,oo)` only
C. `(-oo,0)uu(0,oo)` only
D. `(-oo,0)` only

Answer

A

Explanation

Given `f(x)=(x)/(1+|x|)`
`={{:((x)/(1-x)",",xlt0),((x)/(1+x)",",xge0):}(":'|x|={{:(x if ,xge0),(-x if ,xlt0):}`
`:.LHD=f(0^(-))=underset(hto0)lim(f(0-h)-f(0))/(-h)`
`=underset(hto0)lim(f(-h)-f(0))/(-h)`
`=underset(hto0)lim((h)/(1+h)-0)/(h)=underset(hto0)lim(1)/(1+h)=1`
Since, LHD=RHD
`:.f(x)` is differentiable at x=0
Hence, f(x) is differentiable in `(-oo,oo)`.