# Max/Min Problems with an inequality constraint

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?
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$|z|\le2$ implies $f(z)\le|z^2|+|iz|=|z|^2+|z|\le2^2+2=6$. Since $f(-2i)=6$, there's your answer.
You can apply the Maximum Modulus Principle and conclude that the maximum is on $\vert z\vert = 2$, then apply the Lagrange Multiplier Method. ;)