Let `f:RtoR` be defined as `f(x)=sin(|x|)`
Which one of the following is correct?

Options

A. f is not differentiable only at 0
B. f is differentiable at 0 only
C. f is differentiable everywhere
D. f is non-differentiable at many points

Answer

A

Explanation

Given functions is : `f(x)=sin|x|`
`={{:(sin(x)",",xge0),(sin(-x)",",xlt0):}`
`={{:(sinx",",xge0),(-sinx",",xlt0):}`
`LHD" at "=0underset(hto0)lim(f(0-h)-f(0))/(0-h-0)`
`=underset(hto0)lim(f(0-h)-f(0))/(-h)=underset(hto0)lim(-sin(-h)-0)/(-h)=-1`
`RHD" at "x=0underset(hto0)lim(f(0+h)-f(0))/(0+h-0)`
`=0underset(hto0)lim(f(0+h)-f(0))/(h)=underset(hto0)lim(sin(h-0))/(h)=1`
`LHDneRHD`
f(x) is not differentiable at x=0.