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Discuss the continuity and differentiability of the fx=x+x-1 in the interval -1,2

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Given: fx=x+x-1x=-x for x<0x=x for x>0x-1=-x-1=-x+1 for x-1<0 or x<1x-1=x-1 for x-1>0 or x>1Now,fx=-x-x+1=-2x+1 x-1,0orfx=x-x+1=1 x0,1orfx=x+x-1=2x-1 x1,2
Now,LHL=limx0- fx=limx0- -2x+1=0+1=1RHL=limx0+ fx=limx0+ 1=1Hence, at x=0, LHL=RHLAgain,LHL=limx1- fx=limx1-1=1RHL=limx1+ fx=limx1+ 2x-1=2-1=1Hence, at x=1, LHL=RHL
Now,
fx=-x-x+1=-2x+1 x-1,0f'x=-2 x-1,0orfx=x-x+1=1 x0,1f'x=0 x0,1orfx=x+x-1=2x-1 x1,2f'x=2 x1,2
Now,LHL=limx0- f'x=limx0- -2=-2RHL=limx0+ f'x=limx0+ 0=0Since, at x=0, LHLRHLHence, fx is not differentiable at x=0Again,LHL=limx1- f'x=limx1- 0=0RHL=limx1+ f'x=limx1+ 2=2Since, at x=1, LHLRHLHence, fx is not differentiable at x=1

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