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The minimum value of |2z-1| + |3z-2| is …..........?

The minimum value of |2z-1| + |3z-2| is …..........?

Grade:12th pass

1 Answers

mycroft holmes
272 Points
7 years ago
We can write the expression as 2 \left|z - \frac{1}{2} \right| + 3 \left|z - \frac{2}{3} \right|
 
= 2 \left( \left|z - \frac{1}{2} \right| + \left|z - \frac{2}{3} \right| \right) + \left|z - \frac{2}{3} \right|
which by triangle inequality \ge \left|z - \frac{1}{2} + \frac{2}{3}-z \right| + \left|z - \frac{2}{3} \right|
 
=\frac{1}{6}+ \left|z - \frac{2}{3} \right|
Now, the equality occurs when z is any real numbers that lies in the interval [1/2, 2/3]
 
Its now clear that among these if we set z = 2/3, then we attain the minimum value of \boxed{\frac{1}{6}}

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