Show that the function f defined as follows,

f (x) = rr 3x-2 , & 0<x less than equal to 3 2x^2 - x , & 1<x less than equal to 2 5x-4 , &x>2


Is continuous at x = 2, but not differentiable there at x = 2.

For continuity,


LHl(at x = 2) = RHL (at x = 2)


f(2) = 2(2)2 - 2


= 8 - 2 = 6


LHL =


=


=


= 8 - 2


= 6


RHL =


=


=


= 6


Since, LHL = RHL = f(2)


Hence, F(x) is continuous at x = 2


For differentiability,


LHD(at x = 2) = RHD (at x = 2)


(LHD at x = 2) =


=


=


=


=


=


=


= 6


(RHD at x = 2) =


=


=


=


= 5


Since, (RHD at x = 2)(LHD at x = 2)


Hence, f(2) is not differentiable at x = 2.


4