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JEE Main202325 Jan 2023Morning ShiftMathematicsDefinite IntegrationActual

The minimum value of the function f x = ∫ 0 2 e x - t d t is

Options

  1. A2 e - 1
  2. B2 e - 1
  3. C2
  4. De e - 1

Correct answer

A. 2 e - 1

Step-by-step solution

Given, f x = ∫ 0 2 e x - t d t Now For x ≤ 0 f x = ∫ 0 2 e t - x d t = e - x e 2 - 1 And for 0 < x < 2 f x = ∫ 0 x e x - t d t + ∫ x 2 e t - x d t = e x + e 2 - x - 2 For x ≥ 2 f x = ∫ 0 2 e x - t d t = e x - 2 e 2 - 1 Now for x ≤ 0 ,   f x is decreasing as e - x is decreasing function and x ≥ 2 ,   f x is increasing as e x - 2 is increasing function, So, minimum value of f x lies in x ∈ 0 , 2 Applying A.M ≥ G.M in e x + e 2 - x we

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