Discussion
If z is a complex number, then the minimum value of \(|z-2| + |z-4|\) is
Answer: E
The minimum value is 2.
\(|z - 2|\) and \(|z - 4|\) are the distances of the point z from the points 2 and 4 on the real axis. By the triangle inequality,
\[ |z-2| + |z-4| \ge |(z-2) - (z-4)| = 2 \]
Equality holds for every z on the segment joining 2 and 4, for example z = 3, which gives 1 + 1 = 2. The sum can never be smaller than the distance between the two fixed points, so 2 is the minimum. The values 4, 6, \(\sqrt{6}\) and \(\sqrt{20}\) are all attained at points off that segment, so none is the least.
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