The correct options are
A (−∞,∞)
B [0, ∞)
C (−∞, 0)∪(0, ∞)
D (0, ∞)
f(x)=x1+|x|=(x1+x, if x≥0x1−x, if x<0Looking at f(x), we can say that f(x) is differentiable at every where except 0,where we need to check its differentiability.(LHD at x=0)=limh→0− f(0−h)−f(0)h=limh→0− −h1+h−0−h= limh→0−11 + h=1(RHD at x=0)=limh→0+ f(0+h)−f(0)h=limh→0+ h1+h−0h=limh→0+11 + h=1∵(LHD at x=0)=(RHD at x=0)∴f is differentiable at x=0.Hence, f is differentiable in (−∞,∞)