1. If $\displaystyle f $ is analytic in a closed bounded region $\displaystyle G $ and $\displaystyle f(z) \neq 0 $ in $\displaystyle G $, show

that $\displaystyle |f| $ assumes its minimum value on the boundary of $\displaystyle G $.

Hint: consider $\displaystyle \frac{1}{f} $.

2. Use Problem 1 to prove the Fundamental Theorem of Algebra.