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AP EAMCET202120 Aug 2021Evening ShiftMathematicsApplication of DerivativesActual

If the function f x = 2 x 3 - 9 a x 2 + 12 a 2 x + 1 , where a > 0 , attains its maximum and minimum at p and q respectively such that p 2 = q , then a equals

Options

  1. A0
  2. B1
  3. C2
  4. D- 1

Correct answer

C. 2

Step-by-step solution

Given f ( x ) = 2 x 3 - 9 a x 2 + 12 a 2 x + 1 . Differentiating f x with respect to x , we get f ' x = 6 x 2 - 18 a x + 12 a 2 Again, differentiating f ' x with respect to x , we get f " x = 12 x - 18 a For maximum or minimum, f ' x = 0 . ⇒ 6 x 2 - 18 a x + 12 a 2 = 0 ⇒ x 2 - 3 a x + 2 a 2 = 0 ⇒ x - a x - 2 a = 0 ⇒ x = a   or x = 2 a Now, f " a = 12 a - 18 a = - 6 a < 0 and f " 2 a = 24 a - 18 a = 6 a > 0 . Therefore, f x is maximum at x = a and minimum

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