Let f(z) = u(x, y) +iv(x, y) is continuous on a closed bounded region R and f is analytic and not constant in the interior of R. Prove that the component function u(x, y) has a maximum and minimum values of v(x, y) are reached on the boundary of R and never in the interior.

Scenario: susan wants to make 2 square flags to sell at a crafts fair. the fabric she wants to buy is 3 meters wide. she doesn't want any fabric left over. what's the least amount of fabric she should buy? question: which equation will susan solve her problem? note: let x represent the length of 1 side of the flag. options: 1) 2x^2 = 4x 2) 8 +2x = 2(4x) 3) 2 * 2 = 4 * 2 4) 4x^2 -2x = 0