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

fx= -2x for x<-1  2 for -1x<1  2x for x1LHD=lim x-1-fx-f-1x--1LHD=lim x-1--2x-2x+1LHD=lim x-1--2x-2x+1LHD=lim x-1--2x+1x+1LHD=lim x-1--2 = -2RHD=lim x-1+fx-f-1x--1RHD=lim x-1+2-2x+1 = 0LHD at x = -1  RHD at x = -1,so function is not differentiable at x = -1Similarly, we can prove that given function is not differentiable at x = 1..

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f(x)= -2x for x<-1 , 2 for x belongs to -1 to 1 and 2x for x>1 
at x=-1 f(x-h)=-2(x-h)-(-2x)/-h =-2  and f(x+h ) =2-2/h=0at x=-1  thus RHD doesnt equals LHD thus function is not differentiable at x=-1 similarly it is not differentiable at x=1 . also from graph we see that function has sharp edge at these points thus not differentiable
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