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COMEDK2018MathematicsContinuity and Differentiability

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

(a) We have, f(x)= |x|+|x|= array cc (-x)+(-x), & x < 0 (x)+x, & x 0 array .= cases x-x, & x < 0 x+x, & x 0 cases f^ (x)= cases - x-1, & x < 0 - x+1, & x 0 cases f^ (0⁻ )=-1 and f^ (0⁺ )=1 f^ (0⁻ ) f^ (0⁺ ) So, it is not differentiable at x=0 . (b) We have, aligned f(x) &= |x|-|x| &= array rr (-x)-(-x), & x < 0 (x)-x, & x 0 array . &= cases x+x, & x < 0 x-x, & x 0 cases f^ (x) &= cases - x+1, & x < 0 - x-1, & x 0 cases f^ (0⁻ ) &=1 and f^ (0⁺ )=-1 aligned So, it is not differentiable at x=0 .

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