The Lagrange Multiplier Method is usually used to deal with the maximizing or minimizing problem subject to a constraint which is usually an equality. Consider the following problem:
$f(z)=|z^2-iz|,\quad z\in{\mathbb C}$ where $|z|\leq 2$ What is the maximum of $f(z)$?
It seems that the Lagrange Multiplier can not be used here. What I think is that one may let $z=x+iy$, find the critical point, and use the second-derivative test.
- Is there a quick way to solve this problem?