Q.The feasible region of a linear programming problem is the bounded region in the first quadrant ($x \ge 0$, $y \ge 0$) satisfying $x + 2y \le 76$ and $2x + y \le 104$. Its corner points are $O(0, 0)$, $A(52, 0)$, $E(44, 16)$ and $D(0, 38)$. Determine the maximum value of $Z = 3x + 4y$ over this region.

The Corner Point Theorem: Why the Best Answer Hides at the Edges

Imagine maximising profit for a factory that makes two products, with limited raw materials, machine hours, and labour. Every combination that doesn't break a limit is a feasible solution. Plot them all on a graph and they form a shape — always a polygon if your constraints are straight lines.

Where is the best (maximum profit) point? You might think anywhere inside the shape. But the Corner Point Theorem says something surprising: the best point is always at a corner — a vertex of the polygon. Never floating in the middle of an edge or inside.

The Intuition: Why Corners Win

Think of profit as a line you slide across the polygon; each position represents a profit level. You push the line as far as possible (higher profit) while still touching the polygon. The last point of contact before the line escapes is always a corner.

Why? Because both the profit line and the polygon's edges are straight, and the farthest point in any straight-line direction from a polygon is always a vertex. This holds for any flat-sided shape.

The Precise Statement

Three key parts:

1. "If it has an optimal solution" — sometimes the problem is unbounded (profit increases forever) or infeasible (no point satisfies all constraints). The theorem applies only when a best answer exists.

2. "At least one corner point" — several corners can give the same optimal value. If the profit line is parallel to an edge, every point on that edge is optimal, including both endpoints (corners).

3. "Of the feasible region" — the polygon formed by all constraints. Corners are where two constraint lines intersect.

Why This Matters for Exams

To solve a linear programming problem:

1. Find all corner points (solve pairs of constraint equations).
2. Plug each corner into the objective function.
3. Pick the best value.

The theorem guarantees you haven't missed a better answer hiding in the middle. …