<p>The function \(f(x) = \dfrac{x}{2} + \dfrac{2}{x}\) has a local minimum at</p>
Step-by-Step Solution
Key Concept: Find critical points by setting f'(x) = 0, then use the second derivative test to confirm it's a minimum. For f(x) = x/2 + 2/x, the critical point occurs where the derivative equals zero and f''(x) > 0.
<p><strong>Step 1:</strong> Find the first derivative.</p><p>f(x) = x/2 + 2/x</p><p>f'(x) = 1/2 - 2/x²</p><p><strong>Step 2:</strong> Set f'(x) = 0 to find critical points.</p><p>1/2 - 2/x² = 0</p><p>1/2 = 2/x²</p><p>x² = 4</p><p>x = ±2</p><p><strong>Step 3:</strong> Find the second derivative to verify minimum.</p><p>f''(x) = 4/x³</p><p><strong>Step 4:</strong> Evaluate f''(x) at critical points.</p><p>At x = 2: f''(2) = 4/8 = 1/2 > 0 ✓ (local minimum)</p><p>At x = -2: f''(-2) = 4/(-8) = -1/2 < 0 (local maximum)</p><p><strong>Step 5:</strong> Verify the minimum value.</p><p>f(2) = 2/2 + 2/2 = 1 + 1 = 2</p><p>∴ The function has a local minimum at <strong>x = 2</strong> with minimum value 2</p>
Correct Answer: A