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JEE Advanced2016MathematicsContinuity and DifferentiabilityActual

Let a , b ∈ R and f : R → R be defined by f x = a cos ⁡ x 3 - x + b x sin ⁡ x 3 + x . Then f is -

Options

  1. ADifferentiable at x = 0 if a = 0 and b = 1
  2. BDifferentiable at x = 1 if a = 1 and b = 0
  3. CNOT differentiable at x = 0 if a = 1 and b = 0
  4. DNOT differentiable at x = 1 if a = 1 and b = 1

Correct answer

A. Differentiable at x = 0 if a = 0 and b = 1

Step-by-step solution

If x 3 - x ≥ 0 ⇒ cos ⁡ x 3 - x = cos ⁡ x 3 - x x 3 - x < 0 ⇒ cos ⁡ x 3 - x = cos ⁡ x 3 - x Similarly b x sin ⁡ x 3 + x = b x sin ⁡ x 3 + x for all x ∈ R ∴ f x = a cos ⁡ x 3 - x + b x sin ⁡ x 3 + x Which is composition and sum of differentiable functions therefore always continuous and differentiable.

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