Let `f(x)` be defined on `[-2,2]` and be given by
`f(x)={(-1",",-2 le x le 0),(x-1",",1 lt x le 2):} and g(x)=f(|x|) +|f(x)|`.
Then find `g(x)`.

Options

A. not differentiable at one point
B. not differentiable at two points
C. differentiable at all points
D. not continuous

Answer

A

Explanation

According to given function
`|f(x)|={{:(1,,,-2lexlt0),(|x^2-1|,,,0 le x le 2):}={{:(+1,,,-2 le x lt0),(1-x^2,,,0 le x lt 1),(x^2-1,,,1le x le 2):}`
and `f|x|={{:(-1,,,-2le|x|lt0),(x^2-1|,,,0le|x|le2=0 x in [-2,2]):}`
Hence g(x)={|f(x)|+f(|x|)}=`{{:(x^2,x in(-2,0]),(0, x in (0,1]),(2(x^2-1), x in (1,2)):}`
As `{:(g(0^+)=g(0^-)=g(0)),(g(1^+)=g(1^-)=g(1)):}}rArr ` g(x) is continuous in (-2,2)
`{:(g'(0^+)=g'(0^-)),(g'(1^+)ne g'(1^-)):}} rArr g(x)` is differentiable at x=0 but it is non differentiable at x=1