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JEE Advanced2019MathematicsApplication of DerivativesActual

Let f : R → R be given by f x = x - 1 x - 2 x - 5 . Define F x = ∫ 0 x f t d t , x > 0 . Then which of the following options is/are correct?

Options

  1. AF has a local minimum at x = 1
  2. BF has a local maximum at x = 2
  3. CF x ≠ 0 for all x ∈ 0 , 5
  4. DF has two local maxima and one local minimum in 0 , ∞

Correct answer

A. F has a local minimum at x = 1

Step-by-step solution

F x = ∫ 0 x f t . d t F ' x = f x = x - 1 x - 2 x - 5 ⇒ F x has maxima at x = 2 and minima at x = 1 and x = 5 now F 2 = ∫ 0 2 x 3 - 8 x 2 + 17 x - 10 . d x = x 4 4 - 8 x 3 3 + 17 x 2 2 - 10 x 0 2 F 2 = - 10 3 which is a maxima in x ∈ 0 , 5 F 0 = 0 Hence F x 0 ∀ x ∈ 0 , 5

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