Question

The minimum value of $|z|+|z-1|+|z-2|$ is

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is C $2$

Let $z=x+iy$

$f\left(z\right)=|z|+|z-1|+|z-2|$

$f(x,y)=\sqrt{x^2+y^2}+\sqrt{(x-1{)}^2+y^2}+\sqrt{(x-2{)}^2+y^2}$

For minimum value, $y=0$

$f\left(x\right)=|x|+|x-1|+|x-2|$ for all $x$

The function can be expressed as follows

$f\left(x\right)=3x-3$, $x\ge 2$

$f\left(x\right)=x+1,$ $1\le x<2$

$f\left(x\right)=3-x$, $0\le x\le 1$

$f\left(x\right)=-3x+3$, $x<0$

Hence, $f\left(min\right)=f\left(1\right)=2$