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JEE Main202224 Jun 2022Morning ShiftMathematicsApplication of DerivativesActual

For the function f x = 4 log e x - 1 - 2 x 2 + 4 x + 5 , x > 1 , which one of the following is NOT correct?

Options

  1. Af x is increasing in 1 , 2 and decreasing in 2 , ∞
  2. Bf x = - 1 has exactly two solutions
  3. Cf ' e - f " 2 < 0
  4. Df x = 0 has a root in the interval e , e + 1

Correct answer

C. f ' e - f " 2 < 0

Step-by-step solution

Given f x = 4 log e x - 1 - 2 x 2 + 4 x + 5 , x > 1 f ' x = 4 x - 1 - 4 x - 1 For 1 < x < 2 ⇒ f ' x > 0 For x > 2 ⇒ f ' x < 0 Hence, f x has a maxima at x = 2 f x = - 1 has two solution as the curve of f x will cut it at exactly two points. Now, f e > 0 and f e + 1 < 0 i.e. f e · f e + 1 < 0 f ' e - f " 2 = 4 e - 1 - 4 e - 1 + 8 > 0

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