Q. 53.7( 15 Votes )

Discuss the conti

Answer :

f(x) =


f(x) =


We know that a polynomial and a constant function is continuous and differentiable every where. So, f(x) is continuous and differentiable for x ( - 1,0) and x (0,1) and (1,2).


We need to check continuity and differentiability at x = 0 and x = 1.


Continuity at x = 0


= 1


= 1


F(0) = 1



Since, f(x) is continuous at x = 0


Continuity at x = 1


= 1


= 1


F(1) = 1


= 1


Since, f(x) is continuous at x = 1


For differentiability,


LHD(at x = 0) = RHD (at x = 0)


Differentiability at x = 0


(LHD at x = 0) =


=


=


= 2


(RHD at x = 0) =


=


=


= 0


Since,(LHD at x = 0)(RHD at x = 0)


So, f(x) is differentiable at x = 0.


For differentiability,


LHD(at x = 1) = RHD (at x = 1)


Differentiability at x = 1


(LHD at x = 1) =


=


= 0


(RHD at x = 1) =


=


=


Since, f(x) is not differentiable at x = 1.


So, f(x) is continuous on ( - 1,2) but not differentiable at x = 0, 1


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