Problem 1519 (difficulty: 7/10)

Let \(\displaystyle f(z)\in \C\) be non-constant. Prove the following

(a) \(\displaystyle \re f\) and \(\displaystyle \im f\) have no local extrema.

(b) If \(\displaystyle |f|\) has a local extremum at \(\displaystyle z_0\), then \(\displaystyle f(z_0)=0\).

(c) Prove the fundamental theorem of algebra.

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