Applications of Derivatives
Rolle's Theorem / Mean Value Theorem
Grade 12
Question:
<p>Let \(0 < a < b < \dfrac{\pi}{2}\). If \(f(x) = \begin{vmatrix} \sin x & \sin a & \sin b \\ \cos x & \cos a & \cos b \\ \tan x & \tan a & \tan b \end{vmatrix}\), then the minimum possible number of roots of \(f'(x) = 0\) lying in \((a, b)\) is:</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: Use the AM-GM inequality on the constraint and recognize that the function f(x) = x(π - 2x) is maximized when its derivative equals zero, which occurs at the point where AM-GM achieves equality.
<p><strong>Step 1:</strong> We need to maximize f(x) = x(π - 2x) for x ∈ (0, π/2).</p><p><strong>Step 2:</strong> Take the derivative: f'(x) = π - 2x - 2x = π - 4x.</p><p><strong>Step 3:</strong> Set f'(x) = 0: π - 4x = 0 ⟹ x = π/4.</p><p><strong>Step 4:</strong> Verify this is a maximum using the second derivative test: f''(x) = -4 < 0, confirming a maximum.</p><p><strong>Step 5:</strong> Check that π/4 ∈ (0, π/2) ✓. Alternatively, by AM-GM: x + (π - 2x) ≥ 2√(x(π - 2x)), with equality when x = π - 2x, giving x = π/3. Note: This approach requires careful constraint analysis; the derivative method directly yields x = π/4.</p><p>∴ Answer: B</p>
Correct Answer: B