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JEE Main202329 Jan 2023Evening ShiftMathematicsContinuity and DifferentiabilityActual

Let f and g be twice differentiable functions on R such that f " x = g " x + 6 x f ' 1 = 4 g ' 1 - 3 = 9 f 2 = 3 g 2 = 12 Then which of the following is NOT true ?

Options

  1. Ag - 2 - f - 2 = 20
  2. BIf - 1 < x < 2 , then f x - g x < 8
  3. Cf ' x - g ' x < 6 ⇒ - 1 < x < 1
  4. DThere exists x 0 ∈ 1 , 3 2 such that f x 0 = g x 0

Correct answer

B. If - 1 < x < 2 , then f x - g x < 8

Step-by-step solution

Given functions are: f " x = g " x + 6 x             … 1 f ' 1 = 4 g ' 1 - 3 = 9         … 2 f 2 = 3   g 2 = 12             … 3 By integrating 1 , we get f ' x = g ' x + 3 x 2 + C At x = 1 f ' 1 = g ' 1 + 3 + C ⇒ 9 = 3 + 3 + C ⇒ C = 3 ∴    f ' x = g ' x + 3 x 2 + 3 Again by integrating, we get f x = g x + x 3 + 3 x + D At x = 2 , we get f 2 = g 2 + 8 + 3

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