Show that the function f(x)=|x+1|+|x-1| for all x belongs to R, is not differentiable at points x= -1 and x=1.

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Solution

For Values x<-1 , f(x) = -(x+1) -(x-1)
= -2x
For Values -1 < x < 0
f(x) = x+1 -(x-1)
= 2

Left hand limit not Equals Right hand Limit hence not differentiable at -1

For Values x>1 , f(x) = x+1 + x-1
= 2x
For Values 0<x < 1 , f(x) = x+1 -(x-1)
= 2

Left Hand limit not Equals Right hand Limit hence not differentiable at 1

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