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KVPY2018MathematicsContinuity and Differentiability

Let f x = x |sin x| , x ∈ R . Then,

Options

  1. Af is differentiable for all x , except at x = n π , n = 1 , 2 , 3 . . . . ,
  2. Bf is differentiable for all x , except at = n π , n = ± 1 , ± 2 , ± 3 ,
  3. Cf is differentiable for all x , except at x = n π , n = 0 , 1 , 2 , 3
  4. Df is differentiable for all x , except at x = n π , n = 0 , ± 1 , ± 2 , ± 3 , …

Correct answer

B. f is differentiable for all x , except at = n π , n = ± 1 , ± 2 , ± 3 ,

Step-by-step solution

We have, f x = x | sin x | , x ∈ R f x = x sin x , x ∈ 2 n π , 2 n + 1 π - x sin x , x ∈ 2 n + 1 π , 2 n π f ' n π = lim x → n π f x - f n π x - n π = lim x → n π x | sin x | x - n π Clearly, f x is differentiable for all x except x = n π , n = ± 1 ± 2 , ± 3 , …

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