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JEE Advanced2011MathematicsContinuity and DifferentiabilityActual

If f(x)= array cc -x- 2 , & x - 2 - x, & - 2 1 array . , then

Options

  1. Af(x) is continuous at x=- 2
  2. Bf(x) is not differentiable at x=0
  3. Cf(x) is differentiable at x=1
  4. Df(x) is differentiable at x=- 3 2

Correct answer

A. f(x) is continuous at x=- 2

Step-by-step solution

f(x)= array cc -x- 2 , & x - 2 - x, & - 2 1 array . Continuity at x=- 2 , aligned & f (- 2 )=- (- 2 )- 2 =0 & RHL _ h 0 - (- 2 +h )=0 aligned Continuous at x=0 Continuity at x=0 f(0)=-1 RHL _ h 0 (0+h)-1=-1 Continuous at x=0 Continuity at x=1 ; f(1)=0 RHL _ h 0 (1+h)=0 Continuous at x=1 Here, f^ (x)= array cc -1, & x - 2 x, & - 2 1 array . Differentiable at x=0 , LHD =0, RHD =1 Not differentiable at x=0 Differentiable at x=1 , LHD =1, RHD =1 Differentiable at x=1 Also, for x=- 3 2 f(x)=-x- 3 2 Differentiable at x=-

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