We find the global extrema of $f(x,y)=x^2 - y^2$ on the region $[-2, 1] \times [-1, 2]$.
Interior Critical Points
Boundary 1 West: $(-2, t)$ for $t \in [-1, 2]$
Boundary 2 East: $(1,t)$ for $t \in [-1, 2]$
Boundary 3 South: $(t, -1)$ for $t \in [-2, 1]$
Boundary 4 North: $(t,2)$ for $t \in [-2, 1]$
Global maximum of 4 at (-2, 0).
Global minimum of -4 at (0, 2).