The minimum value of $|\mathrm{z}|+|\mathrm{z}-1|+|\mathrm{z}-2|$ is

Let $z = x+iy$

$f(z) = |z| + |z-1| +|z-2|$

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

For minimum value, $y= 0$

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

The function can be expressed as follows

$f(x) = 3x -3$, $x \geq 2$

$f(x) = x+1,$ $1\leq x<2$

$f(x) = 3- x$, $0\leq x\leq 1$

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

Hence, $f(min) = f(1) = 2$