Question

The minimum value of $f\left(z\right)=|z|+|z-1|+|z+2|$ is

• A. 3
• B. 4
• C. 5
• D. 6

Answer 1

The correct option is A 3

Taking $|z-1|+|z+2|$
We know
$\left|z_1\right|+\left|z_2\right|⩾|z_1+z_2|$

$|z-1|+|z+2|=|-z+1|+|z+2|⩾|1-\text{/}z+\text{/}z+2|$

$⩾3$

And clearly the value 3 occurs

When z=0, -2, 1

Now, $\left|z\right|isminimumatz=0$

Hence minimum value $=$ 3

Similarly

Taking $\left|z\right|+|z-1|$

$\Rightarrow \left|z\right|+|z-1|=\left|z\right|+|1-z|⩾|\text{/}z+1-\text{/}z|$

$=1$

And value 1
occurs at$z=0,z=1orz=z=\frac{1}{2}$

But $|z+2|$ is minimum at $z=0$

Hence minimum value $=i+|z+2|$

$=3atz=0$

Similarly

Taking $\left|z\right|+|z+2|$
$\Rightarrow \left|z\right|+|z+2|=\left|z\right|+|2-z|⩾|\text{/}z-2-\text{/}z|$
$=2$
But value 2 occurs at o, -1 and -2

But $|z-1|$ is minimum at z$=0$

Hence minimum

value $=2+|z-1|$

$=3$