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AP EAMCET202120 Aug 2021Morning ShiftMathematicsApplication of DerivativesActual

Find the positive value of ' a ' for which the equality 2 α + β = 8 holds, where ' α ' and ' β ' are the points of maximum and minimum, respectively, of the function f ( x ) = 2 x 3 - 9 a x 2 + 12 a 2 x + 1

Options

  1. A0
  2. B2
  3. C1
  4. D1 4

Correct answer

B. 2

Step-by-step solution

Given, a > 0 f ( x ) = 2 x 3 - 9 a x 2 + 12 a 2 x + 1 f ' x = 6 x 2 - 18 a x + 12 a 2 f ' x = 6 x 2 - 3 a x + 2 a 2 f ' x = 6 x - 2 a x - a . x = a ,   x = 2 a f ' ' x = 6 2 x - 3 a f ' ' a = 6 2 a - 3 a < 0 f ' ' 2 a = 6 4 a - 3 a > 0 Thus, α = a ,   β = 2 a ⇒ 2 α + β = 8 ⇒ 2 a + 2 a = 8 ⇒ 4 a = 8 ⇒ a = 2

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