Q.Refer to Exercise 32. Maximum of - Minimum of
(A)
(B)
(C)
(D)
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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.
This theorem is the backbone of linear programming — the method for solving optimisation problems with straight-line constraints.
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.
Solving a linear programming problem by hand, you only need to check the corners — usually just 3–5 points, not the infinite points inside.
The Precise Statement
Corner Point Theorem (Fundamental Theorem of Linear Programming):
If a linear programming problem has an optimal solution, then that optimal solution occurs at at least one corner point (vertex) of the feasible region.
Three key parts:
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"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.
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"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).
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"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:
- Find all corner points (solve pairs of constraint equations).
- Plug each corner into the objective function.
- Pick the best value.
The theorem guarantees you haven't missed a better answer hiding in the middle. …
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