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JEE Main202324 Jan 2023Evening ShiftMathematicsFunctionsActual

If f x = x 3 - x 2 f ' 1 + x f " 2 - f ' ' ' 3 , x ∈ R , then

Options

  1. A3 f 1 + f 2 = f 3
  2. Bf 3 - f 2 = f 1
  3. C2 f 0 - f 1 + f 3 = f 2
  4. Df 1 + f 2 + f 3 = f 0

Correct answer

C. 2 f 0 - f 1 + f 3 = f 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 Also, f ' 1 = a ⇒ 3 - 2 a + b = a ⇒ 3 a = b + 3 Also, f " 2 = 12 - 2 a = b ⇒ 12 - 2 a = 3 a - 3 ⇒ 5 a = 15 ⇒ a = 3 ⇒ b = 6 And, f ' ' ' 3 = c ⇒ c = 6 Hence, f x = x 3 - 3 x 2 + 6 x - 6 Now, f 3 + 2 f 0 = 27 - 27 + 18 - 6 - 12 = 0 f 2 + f 1 = 8 - 12 + 12 - 6 + 1 - 3 + 6 - 6 = 0 So, f 3 + 2 f 0 = f 2 + f 1 ⇒ 2 f 0 - f 1 + f 3 = f 2

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