A good way to visualize this problem is to note that the maximum value of cannot be attained at a point inside the polygonal domain; for at that point would have to curve back downward in all directions, and no plane can do this.
The maximum value must therefore be attained at a point on the boundary of the domain. Now, as you move across the boundary lines, either increases or decreases (or stays the same) continually at the same rate due to its planar nature. Thus, we will certainly find the peak value of at a vertex of the domain, as we can move in the increasing direction of the boundary until we find it.
The problem then becomes to find the intersections of the lines
that belong to the boundary of the domain (not all intersections will), and find the one at which is greatest.
Hope this helps.