<p>Given function is \(f(x) = 2x^3 - 9ax^2 + 12a^2x + 1\). If \(f(p) = \max. f(x)\) and \(f(q) = \min. f(x)\), and \(p^2 = q\), then the value of \(a\) is:</p>
Step-by-Step Solution
Key Concept: Find critical points using f'(x) = 0, identify which is maximum (p) and minimum (q), then apply the constraint p² = q to determine a.
<p><strong>Step 1:</strong> Find the derivative: f'(x) = 6x² - 18ax + 12a²</p><p><strong>Step 2:</strong> Set f'(x) = 0: 6x² - 18ax + 12a² = 0 → x² - 3ax + 2a² = 0</p><p><strong>Step 3:</strong> Factor: (x - a)(x - 2a) = 0, so critical points are x = a and x = 2a</p><p><strong>Step 4:</strong> Check second derivative: f''(x) = 12x - 18a. At x = a: f''(a) = 12a - 18a = -6a. At x = 2a: f''(2a) = 24a - 18a = 6a</p><p><strong>Step 5:</strong> For a > 0: f''(a) < 0 (maximum at p = a) and f''(2a) > 0 (minimum at q = 2a)</p><p><strong>Step 6:</strong> Apply constraint p² = q: a² = 2a → a² - 2a = 0 → a(a - 2) = 0</p><p><strong>Step 7:</strong> Since a ≠ 0 (otherwise both critical points coincide), we get a = 2</p><p><strong>Verification:</strong> When a = 2: p = 2 and q = 4, and p² = 4 = q ✓</p><p>∴ Answer: <strong>a = 2</strong></p>
Correct Answer: C