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BITSAT2014MathematicsContinuity and DifferentiabilityActual

Let f(x)= array c (x-1) 1 x-1 if x 1 0 if x=1 array . Then which one of the following is true?

Options

  1. Af is differentiable at x=0 and x=1
  2. Bf is differentiable at x=0 but not at x=1
  3. Cf is differentiable at x=1 but not at x=0
  4. Df is neither differentiable at x=0 nor at x=1

Correct answer

B. f is differentiable at x=0 but not at x=1

Step-by-step solution

We have; f(x)= array ll (x-1) ( 1 x-1 ) & if x 1 0 & if x=1 array .R f^ (1)= _ h 0 f(1+h)-f(1) h = _ h 0 h 1 h -0 h = _ h 0 1 h which does not exist. f is not differentiable at x=1 Also .f^ (0)= 1 (x-1) - x-1 (x-1)² ( 1 x-1 ) ]_ x=0 =- 1+ 1 f is differentiable at x=0

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