Q.State whether the following statement is True or False: Maximum value of the objective function $Z = ax + by$ in a LPP always occurs at only one corner point of the feasible region.

By the Corner Point Theorem an optimum of $Z = ax + by$ occurs at a corner of the feasible region — but not necessarily at only one.

If the objective line $ax + by = k$ is parallel to a boundary edge, its last contact with the region is that entire edge, so both corner endpoints (and every point between) give the same maximum. …

The Corner Point Theorem

The theorem says at least one corner, not exactly one.

When more than one corner is optimal

The objective $Z = ax + by$ has level lines $ax + by = k$, all parallel. Slide the line in the direction of increasing $Z$; the maximum is its last contact with the region. That last contact is either:

• a single vertex (the usual case), or
• an entire edge, when the objective line is parallel to that boundary edge.

In the second case every point on the edge gives the same maximum — infinitely many optimal points, including two corner points, not one.

Example …

Method: Testing a "universal" claim about an LPP optimum

For a True/False statement that asserts something always holds in every linear programming problem, you do not prove it — you test whether one valid LPP can break it.

Steps

Step 1: Read the quantifier exactly.

Words like always and only one carry the whole question. State the Corner Point Theorem in its precise form: if an optimum exists it occurs at at least one corner point — never "exactly one".

Step 2: Look for the parallel-edge (multiple-optima) escape.

A "single optimum" claim fails whenever the objective line $ax+by=k$ is parallel to a boundary edge of the feasible region. Sliding the line outward, its last contact is that whole edge, so every point on it — both endpoints and all points between — is optimal. …

Common Mistakes

Mistake 1: Reading the Corner Point Theorem as "exactly one" corner.

Why it's wrong: the theorem only guarantees an optimum at at least one vertex. Correct approach: remember that when the objective line is parallel to an edge, the whole edge (two corners plus everything between) is optimal, so the maximum can occur at many points.

Mistake 2: Confusing multiple optimal solutions with unboundedness. …