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I am trying to find a simple method that does not use the tools of advanced differential calculus to find following maximum, whose existence is justified by the compactness of the close ball $\Delta$ of $\mathbb C$ and continuity of the function $f:z \mapsto |z^3 + 2iz|$ from $\mathbb C$ to $\mathbb C$

$$ \large { \displaystyle \max_{z \in {\mathbb C},|z| \leq 1} |z^3 +2i z |} $$ Since : $$(\forall z \in \Delta) \quad f(z) \leq 3 $$ is obtained using triangular inequality, we can yet try to find some $z_0 \in {\mathbb C}$ such that $f(z_0)=3$

Does anybody have an idea?

Thanks.

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1  
Why should I? $ $ –  user641 Jun 9 '12 at 12:02
    
Hello Mohamed, welcome to the site. It is more useful to you and helpful to us if you include your thoughts on the problem, as well as your attempts to solve it. –  Paul Slevin Jun 9 '12 at 12:04
    
And, if it's homework, it's also helpful if you add the homework tag. –  Gerry Myerson Jun 9 '12 at 12:18
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Welcome to math.SE, Mohamed: since you are a new user, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are so far; this will prevent people from telling you things you already know, and help them write their answers at an appropriate level. If this is homework, please add the [homework] tag; people will still help, so don't worry. Also, many find the use of imperative ("Find") to be rude when asking for help; please consider rewriting your post –  draks ... Jun 9 '12 at 12:22
    
That's a much better problem statement. It tells us that you have worked quite a bit on the problem, and already know a lot about it. Maybe with my hint you can polish it off. –  Gerry Myerson Jun 9 '12 at 13:04

3 Answers 3

Hint: when do you get equality in the triangle inequality?

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@ Gerry Myerson : When the two terms are directely proportional. I well see this . –  Mohamed Jun 9 '12 at 13:13
    
@ Gerry Myereson: Thank's for hint. Using it, i find that $z_0=\frac{\sqrt 2}{2} (1+i)$ gives $f(z_0)=3$ –  Mohamed Jun 9 '12 at 13:27
    
Well done! ${}$ –  Gerry Myerson Jun 9 '12 at 23:56

If you've begun a study of complex functions, you may have seen the Maximum Modulus Principle. Since $z^3 + 2iz$ is a polynomial and entire (analytic in the complex plane), the maximum of $|z^3 + 2iz|$ you seek must occur on the boundary of the unit disk. Gerry's Hint then quickly points you in the right direction!

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Ah, drat, you type faster +1 –  dbaupp Jun 9 '12 at 13:28

Hint: $z^3 + 2iz$ is differentiable on $\mathbb{C}$ (i.e. holomorphic) so you can apply the maximum modulus principle, and deduce that the maximum of $f$ lies on the boundary of $\Delta$, which has a simple parametrisation, so you can use standard techniques from real one-variable calculus to find the maximum.

(A slightly different approach to Gerry Myerson's, much more complicated in this case but also far more general.)

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@ hardmath and dbaupp : Thank you for the idea of maximum module principle.I was looking for a suitable method for beginning students. It turns out that the Pricipe gives the idea already! Thank you! –  Mohamed Jun 9 '12 at 13:37

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