Why corners are enough

A corner point (vertex) of the feasible region is a point of the region where two boundary lines cross.

Theorem 1: if Z = ax + by has an optimal value (maximum or minimum) on the feasible region R, that value occurs at a corner point of R.

Theorem 2: if R is bounded, Z has both a maximum and a minimum on R, and each occurs at a corner point.

If R is unbounded, a maximum or a minimum may fail to exist; but if one does exist, Theorem 1 still puts it at a corner.

The picture behind the theorems: the points where Z takes a fixed value k lie on the straight line ax + by = k. Changing k slides this line parallel to itself, and the last point of R it touches on the way out is a corner (or a whole edge through two corners).

The proofs are beyond the textbook; the theorems are used as tools.