Min. value

**jacks** · January 10th 2011, 08:06 PM

If $Z$ is a Complex no. . Then Find Min. value of $|2Z-1|+|3Z-2|$

**Failure** · January 10th 2011, 11:47 PM

Originally Posted by jacks

If $Z$ is a Complex no. . Then Find Min. value of $|2Z-1|+|3Z-2|$

Consider that $|2z-1|+|3z-2|=2\cdot |z-1/2|+3\cdot |z-2/3|$ . Now, surely, the value of z that minimizes this expression lies on the line between $z_1:= 1/2$ and $z_2 := 2/3$ and is, therefore, a real number.
In fact, it must be $z = 2/3$ , for if you shift a $z\in\mathbb{R}$ that lies between $z_1$ and $z_2$ a little closer towards $z_2$ , without exceeding it, the first term of the sum gets larger but the second term gets smaller by a larger amount. This continues until you reach $z_2$ ; after that point on the real axis both terms increase.

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