Minimum of p(z)

[man111]

Let $P(z) = \left|2z-1-i\right|+\left|3z-2-2i\right|+\left|4z-3-3i\right|$ then find Minimum value of $P(z)$ also find corrosponding complex number $z$.

[myra_panama]

first of all let $z=x+iy$ , then use $|z|=\sqrt{x^2+y^2}$ ,,,, maybe something you'll get

[Руст]

Let $z=x(1+i)$, then $f(x)=\sqrt 2 (|2x-1|+|3x-2|+|4x-3|)$. Obviosly minimum, when x is real $\frac{1}{2}<x<\frac{3}{4}$.
It is easy to chek, that $f(x)$ degrease for $\frac 12 <x<\frac 23$ and increase for $x>\frac 23$. Therefore minimum, when $x=\frac 23$ or $z=\frac{2+2i}{3}$ and f_{min}$=\frac{\sqrt 8}{3}.$

[мат-ламер]

Ясен пень (obviously), что минимум взвешенной суммы растояний (minimum summa of weigted distances) до трёх точек, лежащих на одной прямой (from three points lying on the line) достигается на одной из них (is achived on one of them).

[man111]

thank pyctr and мат-ламер myra