Question

Let \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \), where \( a > 0 \). The minimum of \( f \) is attained at a point \( q \) and the maximum is attained at a point \( p \). If \( p^3 = q \), then \( a \) is equal to:

• A. \( 1 \)
• B. \( 3 \)
• C. \( 2 \)
• D. \( \sqrt{2} \)
• E. \( \frac{1}{2} \)

Hint:

For cubic functions, critical points come from \( f'(x)=0 \). The smaller root gives maximum and larger gives minimum when leading coefficient is positive.

Answer 1

The Correct Option is C