Minimum value of a complex function

Question

What is the minimum value of $|z+2+i|+|z-2-i|+|3-z|+|2-z|?\;\;$ $2, 3, 5, 6$

I was trying to solve this problem by assuming $z= r(\cos x+i\sin x)$ but then the calculations become lengthy and complicated. Can someone help me please?

Answer 1

Providing this is a multiple choice question

We have

$$|z+2+i|+|z-2-i|+|3-z|+|2-z|\ge |z+2+i|+|3-z|.$$

The minimum of the RHS is attained in particular when $z$ is at the middle of $-2-i$ and $3,$ and its value is equal to the distance between those two complex numbers, which is equal to $\sqrt {26} >5$.

Hence the only possible answer is $6$.