TITLE: Find a maximum of complex function
QUESTION [7 upvotes]: 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.

REPLY [5 votes]: 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!

REPLY [3 votes]: 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.)