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JEE Main201910 Jan 2019Morning ShiftMathematicsDifferentiationActual

Let, f : R → R be a function such that f x = x 3 + x 2 f ' 1 + x f '' 2 + f ''' 3 , ∀ x ∈ R . Then f 2 equals

Options

  1. A30
  2. B8
  3. C- 4
  4. D- 2

Correct answer

D. - 2

Step-by-step solution

Let, f x = x 3 + a x 2 + b x + c ⇒ f ' x = 3 x 2 + 2 a x + b ⇒ f '' x = 6 x + 2 a ⇒ f ''' x = 6 According to the question, a = f ' 1 = 3 + 2 a + b ⇒ a + b = - 3   . . . i b = f '' 2 = 12 + 2 a ⇒ 2 a - b = - 12   . . . i i c = f ''' 3 ⇒ c = 6 Solving equations i   &   i i , we get, a =   - 5   &   b = 2 ⇒ f x = x 3 - 5 x 2 + 2 x + 6 ∴   f 2 = 8 - 20 + 4 + 6 = - 2 .

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