Question

If the function \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \), where \( a>0 \), attains its local maximum and minimum at \( p \) and \( q \), respectively, such that \( p^2 = q \), then \( f(3) \) is equal to:

• A. \) is equal to: \vspace{0.2cm}
• B. 55
• C. 10
• D. 23

Hint:

Critical points from \( f'(x) = 0 \) help determine max/min values. Use given condition on them.

Answer 1

The Correct Option is D

First derivative: \[ f'(x) = 6x^2 - 18ax + 12a^2 = 6(x^2 - 3ax + 2a^2) \] Roots of \( f'(x) = 0 \) are \( x = a, 2a \). Given: \( p^2 = q \Rightarrow a^2 = 2a \Rightarrow a = 2 \) Now, \[ f(3) = 2(3)^3 - 9(2)(3)^2 + 12(2)^2(3) + 1 = 54 - 162 + 144 + 1 = 37 \]