Q.

The function f(x)=|x−3|,x≥1x24−3x2+134,x<1

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a

discontinuous at  x = 1

b

not differentiable at x = 1

c

continuous and differentiable at x = 1

d

continuous at x = 1 but not derivable at x  = 1 .

answer is C.

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Detailed Solution

f'1−    =limx→1−f(x)−f(1)x−1                    =limx→1x24−3x2+134−2x−1                   = limx→1x2−6x+54(x−1)                  =limx→1(x−5)(x−1)4(x−1)=limx→1(x−5)4=−1f'1+=limx→1+f(x)−f(1)x−1limx→13−x−2x−1=−1∴f'(1)=−1∴f(x) is derivable and hence continuous also at x=1
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