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


4 years ago

mycroft holmes
272 Points
							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 $\boxed{\frac{1}{6}}$

4 years ago
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