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JEE Main2016MathematicsContinuity and DifferentiabilityActual

If the function f x = - x , x < 1 a + cos - 1 ⁡ x + b , 1 ≤ x ≤ 2 is differentiable at x = 1 , then a b is equal to

Options

  1. Aπ + 2 2
  2. Bπ - 2 2
  3. C- π - 2 2
  4. D- 1 - cos - 1 ⁡ ( 2 )

Correct answer

A. π + 2 2

Step-by-step solution

f x =   - x x < 1 a + cos - 1 ⁡ x + b 1   ≤ x   ≤ 2 f x is continuous at x = 1 as f x is differentiable at x = 1 . ⇒ lim x → 1 f ( x ) = lim x → 1 + ( a + cos − 1 ( x + b ) ) = f ( 1 ) ⇒ - 1 = a + cos - 1 ⁡ 1 +b cos - 1 ⁡ 1 + b = - 1 - a ........ i f x is differentiable at x = 1 . ⇒ LHD = RHD ⇒ - 1 = - 1 1 - 1 + b 2 ⇒ 1 - 1 + b 2 = 1 ⇒ b =   - 1 ...... ii From i ⇒ cos - 1 ⁡ ( 0 ) =   - 1 - a

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