Global Extrema and Boundary Analysis

Closed bounded regions guarantee champions

The boundary is a one-variable problem in disguise

The complete recipe on a worked example

Why the recipe is complete

Which condition guarantees that a continuous function attains an absolute maximum and minimum on a region?

On the rim of the disk $x^2 + y^2 \le 4$, parametrized as $(2\cos t, 2\sin t)$, what does $f(x, y) = x^2 + y^2 - 2x$ become?

With candidates $(1,0)$ at value $-1$, $(2,0)$ at value $0$ and $(-2,0)$ at value $8$ for $f = x^2 + y^2 - 2x$ on the disk of radius $2$, what are the absolute extrema?

The rectangle $0 \le x \le 2$, $0 \le y \le 1$ has its top edge at $y = 1$. Restricting $f(x, y) = x + 2y$ to that edge gives a one-variable function $g(x) = x + 2$. Give the maximum value of $g$ on the edge, as a plain number.

A triangular region's boundary consists of $e$ edges and $c$ corner points. Give $e + c$, the number of boundary sweeps plus corner candidates, as a plain number.