Find a maximum of complex function

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

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!

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.)