<p>If \(z\) is a complex number, then find the minimum value of \(|z| + |z - 1| + |2z - 3|\).</p>
Step-by-Step Solution
Key Concept: Interpret |z| + |z - 1| + |2z - 3| geometrically as sum of distances from point z to fixed points 0, 1, and 3/2 on the complex plane. The minimum occurs when z lies on the line segment connecting these collinear points, specifically at the weighted median position.
<p><strong>Step 1:</strong> Rewrite the expression by recognizing the geometric meaning:</p><p>Let f(z) = |z - 0| + |z - 1| + |2z - 3|</p><p>Note that |2z - 3| = 2|z - 3/2|</p><p>So f(z) = |z - 0| + |z - 1| + 2|z - 3/2|</p><p><strong>Step 2:</strong> Identify the three reference points on the real axis: 0, 1, and 3/2. These represent distances from z to these points (with the third point weighted by factor 2).</p><p><strong>Step 3:</strong> For points on the real axis, let z = x (real number). Then:</p><p>f(x) = |x| + |x - 1| + 2|x - 3/2|</p><p><strong>Step 4:</strong> Analyze by regions. The critical points are x = 0, 1, 3/2.</p><p>For x ∈ [1, 3/2]:</p><p>f(x) = x + (x - 1) + 2(3/2 - x) = x + x - 1 + 3 - 2x = 2</p><p><strong>Step 5:</strong> Check boundary values and other regions:</p><p>• At x = 1: f(1) = 1 + 0 + 2(1/2) = 2</p><p>• At x = 3/2: f(3/2) = 3/2 + 1/2 + 0 = 2</p><p>• For x < 0: f(x) = -x + (1-x) + 2(3/2-x) = 4 - 4x > 2</p><p>• For x > 3/2: f(x) = x + (x-1) + 2(x-3/2) = 4x - 4 > 2</p><p><strong>Step 6:</strong> The minimum value is constant on [1, 3/2] and increases outside this interval.</p><p>∴ Answer: 2</p>
Correct Answer: 2