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JEE Advanced2012MathematicsContinuity and DifferentiabilityActual

Let (x)= array r x² | x |, x 0 0, x=0 array , x R . then is

Options

  1. Adifferentiable both at x=0 and at x=2
  2. Bdifferentiable at x=0 but not differentiable at x=2
  3. Cnot differentiable at x=0 but differentiable at x=2
  4. Ddifferentiable neither at x=0 nor at x=2

Correct answer

B. differentiable at x=0 but not differentiable at x=2

Step-by-step solution

f^ (0⁺ )= _ h 0 f(0+h)-f(0) h aligned = _ h 0 h² | h | h &= _ h 0 h | h | &=0 some finite value =0 aligned array c =0 some finite value =0 and f^ (0⁻ )= _ h 0 f(0-h)-f(0) -h = _ h 0 h² | -h | -h = _ h 0 -h | h |=0 some finite value =0 array array l f^ (0⁺ )=f^ (0⁻ ) f is differentiable at x=0 Now f^ (2⁺ )= _ h 0 f(2+h)-f(2) h = _ h 0 (2+h)² | 2+h |-4 | 2 | h array = _ h 0 (2+h)² ( 2+h ) h = _ h 0 (2+h)² h ( 2 - 2+h )= _ h 0 (2+h)² h ( h 2(2+h) )= _ h 0 (2+h)² h ( h 2(2+h) ) ( h 2(2+h) ) h 2(2+h) = and f^ (2⁻ )= _ h

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