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

Concept: Maximum and minimum points occur where \( f'(x) = 0 \). Given relation \( p^3 = q \), we use critical points to form an equation in \( a \).

Step 1: Find the first derivative.
\[ f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \] \[ f'(x) = 6x^2 - 18ax + 12a^2 \]

Step 2: Find critical points.
\[ 6x^2 - 18ax + 12a^2 = 0 \] Divide by 6: \[ x^2 - 3ax + 2a^2 = 0 \] Factor: \[ (x - a)(x - 2a) = 0 \] \[ \Rightarrow x = a,\; x = 2a \]

Step 3: Identify maximum and minimum points.
For a cubic with positive leading coefficient: - Smaller root \( \Rightarrow \) maximum - Larger root \( \Rightarrow \) minimum \[ p = a,\quad q = 2a \]

Step 4: Use given condition.
\[ p^3 = q \] \[ a^3 = 2a \]

Step 5: Solve for \( a \).
\[ a^3 - 2a = 0 \] \[ a(a^2 - 2) = 0 \] Since \( a > 0 \): \[ a = \sqrt{2} \]

Step 6: Final answer.
\[ \boxed{\sqrt{2}} \]