Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?
share|cite|improve this question
up vote 5 down vote accepted

$|z|\le2$ implies $f(z)\le|z^2|+|iz|=|z|^2+|z|\le2^2+2=6$. Since $f(-2i)=6$, there's your answer.

share|cite|improve this 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. ;)

share|cite|improve this answer
    
Hmm, this gives a general method to solve this kinds of problems. – Jack Oct 14 '11 at 2:55

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.