The correct option is A (−∞,∞)
f(x)=x1+|x| is differentiable everywhere except probably at x = 0.
For x = 0,
(LHD at x=0)=limh→0 f(0−h)−f(0)−h=limh→0 −h1+h−0−h=1(RHD at x=0)=limh→0 f(0+h)−f(0)h=limh→0 h1+h−0h=1(LHD at x=0)=(RHD at x=0)
Therefore, f is differentiable at x = 0.
Hence, f is differentiable in (−∞,∞)