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KVPY2018MathematicsContinuity and Differentiability

f : - 1 , 1 → R be a function defined by f x = x 2 cos π x   for   x ≠ 0 0   for   x = 0 The set of points where f is not differentiable is

Options

  1. Ax ∈ [ - 1 , 1 ] : x ≠ 0
  2. Bx ∈ - 1 , 1 : x = 0 or x = 2 2 n + 1 , n ∈ Z
  3. Cx ∈ - 1 , 1 : x = 2 2 n + 1 , n ∈ Z
  4. D- 1 , 1

Correct answer

C. x ∈ - 1 , 1 : x = 2 2 n + 1 , n ∈ Z

Step-by-step solution

We have, f x = x 2 cos π x   for   x ≠ 0 0   for   x = 0 f x is not differentiable at cos π x = 0 cos π x = 0 ⇒ π x = 2 n + 1 π 2 ⇒ x = 2 2 n + 1 ∵ f x is not differentiable at x ∈ - 1 , 1 : x = 2 2 n + 1 , n ∈ Z

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