If the function f(x) = 2x^3 - 9ax^2 + 12a^2 x + 1 , where a > 0 , attains its maximum and minimum at p and q respectively such that p^2 = q , then a e - Mathematics MCQ AIEEE-CBSE-ENG-03 2026 | ExamTest.live

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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:

Solution & Step-by-step Explanation

Find critical points: f^{'} (x) = 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 Critical points are x = a and x = 2 a . f^{''} (x) = 12 x - 18 a .At x = a : f^{''} (a) = 12 a - 18 a = - 6 a < 0 (Max). So p = a .At x = 2 a : f^{''} (2 a) = 24 a - 18 a = 6 a > 0 (Min). So q = 2 a .Given p^{2} = q ⟹ a^{2} = 2 a .Since a > 0, a = 2 .

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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:

3

1

2

\frac{1}{2}

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