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What will be the minimum value of |2z-1| + |3z-2| ? (A question from complex numbers)

What will be the minimum value of |2z-1| + |3z-2| ? (A question from complex numbers)

Grade:11

1 Answers

jagdish singh singh
173 Points
5 years ago
\hspace{-0.7 cm}$ Let $f(z)=|2z-1|+|3z-2| = 2\left|z-\frac{1}{2}\right|+3\left|z-\frac{2}{3}\right|$\\\\\\$f(z)=2\left(\left|z-\frac{1}{2}\right|+\left|\frac{2}{3}-z\right|\right)+\left|z-\frac{2}{3}\right|\geq \left|z-\frac{1}{2}+\frac{2}{3}-z\right|+\left|z-\frac{2}{3}\right|$\\\\\\So $f(z) = 2\left(\left|z-\frac{1}{2}\right|+\left|\frac{2}{3}-z\right|\right)+\left|z-\frac{2}{3}\right|\geq \frac{1}{6}\;,$when $z=\frac{2}{3}

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