If $|z-i| \leq 4$, find the maximum value of $|iz+13-5i|$
It's easy, the maximum value is $17$.
But, I'm more concerned with the minimum value,
$$|iz+13-5i| \geq |z+i|-|12-5i|$$
Least value of $|z+i|$ can be $0$ and $|12-5i| = 13$
It means the least value of $|iz+13-5i|$ is $-13$? How is this possible?
Best Answer
We have $5\geq 4$, but the smallest value that $5$ can take clearly isn't $4$. The smallest value that $5$ can take is $5$.
Just because your triangle inequality coupled with a lower bound on each term gives you a lower bound of $-13$ for $|iz+13-5i|$, that doesn't mean that $-13$ must be possible to reach.