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BITSAT2010MathematicsQuadratic EquationActual

Which of the following functions is differentiable at (x=0 ) ?

Options

  1. A( (|x|)+|x| )
  2. B( (|x|)-|x| )
  3. C( (|x|)+|x| )
  4. D( (|x|)-|x| )

Correct answer

D. ( (|x|)-|x| )

Step-by-step solution

(|x| ) is non-differentiable function at ( aligned & x=0 as L.H.D =-1 and R.H. D =1 & |x|= array cl x, & x 0 -x, & x < 0 array . aligned ) But ( |h| ) is differentiable ( ) Any combination of two such functions will be non-differentiable. Hence option (a) and (b) are ruled out. Now, consider ( |x|+|x| ) ( aligned & L^ = _ h 0 |-h|+|-h| -h = _ h 0 h -h -1=-1-1 =-2 & R^ = _ h 0 |h|+|h| h = _ h 0 h h +1=1+1=2 & Consider |x|-|x| & L^ = _ h 0 |-h|-|-h| -h = _ h 0 h -h +1=0 & aligned )

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