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JEE Main202324 Jan 2023Morning ShiftMathematicsContinuity and DifferentiabilityActual

Let f x = x 2 sin 1 x ; x ≠ 0 0 ; x = 0 , then at x = 0

Options

  1. Af is continuous but not differentiable
  2. Bf is continuous but f ' is not continuous
  3. Cboth f and f ' are continuous
  4. Df ' is continuous but not differentiable

Correct answer

B. f is continuous but f ' is not continuous

Step-by-step solution

Given: f x = x 2 sin 1 x ;     x ≠ 0 0 ;                                 x = 0 lim x → 0 f x = lim x → 0 x 2 sin 1 x ⇒ lim x → 0 f x = lim x → 0 x 2 × number   between   - 1   to   1 ⇒ lim x → 0 f x = 0 = f 0 Hence, f x is continuous at x = 0 . Now, RHD = f ' 0 + = lim h → 0 f 0 + h - f 0 h ⇒ RHD = f ' 0 + = lim h → 0 h 2 sin 1 h

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