Let f(x) be difined in the interval [-2, 2] such that
`f(x)={{:(-1", " -2lexle0),(x-1", " 0ltxle2):}`
and `g(x)=f(|x|)+|f(x)|`.
Test the differentiability of g (x) in (-2, 2).

Answer

g(x) is differentiable for all ` x in (-2, 2) - {0, 1)`

Explanation

Given that, `f(x) = {{:(-1", " -2 le x le 0),((x-1)", " 0 lt x le 2):}`
Since, `x in [-2, 2]`. Therefore , `|x| in[0, 2]`
`rArr" " f(|x|) = |x|-1, AA x in [-2, 2]`
`rArr" " f(|x|) = {{:(x-1", " 0 le x le 2),(-x-1", "-2 le xle 0):}`
Also, `|f(x)|={{:(1", " -2le xlt 0),(1-x", "0 le x lt 1),(x-1", " 1le x le 2):}`
Also, ` g(x) = f(|x|) + |f(x)|`
`={{:(-x-1+1", " -2 le x le 0),(x-1+ 1 - x", " 0 le x lt 1 ),(x - 1 + x - 1", " 1 le x le 2 ):}`
`g (x) = {{:(-x", " -2 le x le 0),(" 0, " 0 le x lt 1),(2(x-1)"," 1 le x le 2):}`
`:." " g'(x) ={{:(-1", " -2le xle 0),(" 0, " 0 le x lt 1),(2", " 1 le x le 2):}`
`:. ` RHD (at x = 1 ) = 2, LHD (at x = 1) = 0
`rArr ` g(x) is not differentiable at x = 1.
Also, RHD (at x = 0 ) = 0, LHD at (x=0) =- 1
`rArr ` g (x) is not differentiable at x =0 .
Hence, g(x) is differentiable for all ` x in (-2, 2) - {0, 1} `