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BITSAT2016MathematicsContinuity and DifferentiabilityActual

If f(x)= array cl x x (1+x² ) & , x 0 0 & , x=0 array . then f(x) is

Options

  1. Acontinuous as well as differentiable at x =0
  2. Bcontinuous but not differentiable at x=0
  3. Cdifferentiable but not continuous at x =0
  4. Dneither continuous nor differentiable at x =0

Correct answer

A. continuous as well as differentiable at x =0

Step-by-step solution

We have, aligned Lf ^ (0) &= _ h 0 f (0- h )- f (0) - h = _ h 0 - h - h (1+ h ² ) =& _ h 0 (1+ h ² ) ( 0 0 form ) aligned aligned &= _ h 0 - h 2 ~h / (1+ h ² ) =-1 / 2 Rf ^ (0) &= _ h 0 f (0+ h )- f (0) h = _ h 0 h h (1+ h ² ) &= _ h 0 h (1+ h ² ) ( 0 0 form ) &= _ h 0 - h 2 ~h / (1+ h ² ) = -1 2 aligned Since Lf ^ (0)= Rf ^ (0), therefore f ( x ) is dif- ferentiable at x=0 Since differentiability continuity, there-fore f ( x ) is continuous at x =0 .

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