<p>Find the minimum value of \(|z-2|+|z-4|+|z+1-i\sqrt{3}|\)</p>
Step-by-Step Solution
Key Concept: The minimum of sum of distances from a point z to fixed points occurs when z lies on the line segment connecting the two closest points (for two distances), then we add the distance to the third point. Use geometric interpretation: minimize |z-2| + |z-4| first (minimum is 2 when z ∈ [2,4]), then find which point minimizes distance to the third point.
<p><strong>Step 1:</strong> Apply triangle inequality to the first two distances.</p><p>For any complex number z: |z-2| + |z-4| ≥ |2-4| = 2</p><p>Equality holds when z lies on the line segment from 2 to 4 (on the real axis from x=2 to x=4).</p><p><strong>Step 2:</strong> Minimize the sum by choosing z on [2,4].</p><p>The expression becomes: 2 + |z+1-i√3| where z ∈ [2,4] (z is real)</p><p><strong>Step 3:</strong> Find the point z = x ∈ [2,4] that minimizes |x+1-i√3|.</p><p>For z = x (real): |x+1-i√3| = √[(x+1)² + 3]</p><p>This is minimized when d/dx[(x+1)² + 3] = 0</p><p>2(x+1) = 0 → x = -1</p><p><strong>Step 4:</strong> Since x = -1 ∉ [2,4], the minimum on [2,4] occurs at the endpoint closest to -1, which is x = 2.</p><p>At z = 2: |2+1-i√3| = |3-i√3| = √(9+3) = √12 = 2√3</p><p><strong>Step 5:</strong> Calculate the total minimum.</p><p>Minimum value = 2 + 2√3</p><p>∴ Answer: A (which should be 2 + 2√3)</p>
Correct Answer: A