Take the 2-minute tour ×
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.

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|improve this question
add comment

2 Answers

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|improve this answer
add comment

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

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.