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Black Book Advanced Problems in MathematicsMathematicsApplication of Derivatives

Let f:[0, 8] R be differentiable function such that f(0) = 0 , f(4) = 1 , f(8) = 1 , then which of the following hold(s) good ?

Options

  1. AThere exist some c₁ (0, 8) where f'(c₁) = 1 4
  2. BThere exist some c (0, 8) where f'(c) = 1 12
  3. CThere exist c₁, c₂ [0, 8] where 8f'(c₁)f(c₂) = 1
  4. DThere exist some , (0, 2) such that ₀^8 f(t) dt = 3( ^2 f( ^3) + ^2 f( ^3))

Correct answer

A. There exist some c₁ (0, 8) where f'(c₁) = 1 4

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