A corner point of a feasible region is a point in the region which is the _____ of two boundary lines.

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Q. A corner point of a feasible region is a point in the region which is the _________ of two boundary lines.

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Concept understanding — Corner Point Theorem

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.

Note

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.

Tip

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:

"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.
"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).
"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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The Corner Point Theorem tells us that optimal solutions to linear programming problems occur at vertices of the feasible region. A corner point is the intersection of two boundary lines.

The key idea here is simple but powerful. In linear programming, the feasible region is a polygon (or polyhedron in higher dimensions). Its corners — the points where the boundary lines meet — are special. The Corner Point Theorem states that if a linear programming problem has an optimal solution, that solution will occur at one of these corner points. So the blank in your question is asking for the geometric relationship between a corner point and the boundary lines that define it.

Let's think about what "corner point" means visually. Imagine the feasible region as a shape drawn by several straight lines. A corner is where two of those lines cross. That crossing point is shared by both lines — it belongs to each line simultaneously. In mathematical language, we say the point is the intersection of those two boundary lines.

Watch out

A common mistake is to say "meeting point" or "crossing point" — while these are descriptively correct, the precise mathematical term expected in exam contexts is intersection. Be exact with terminology. …

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Q.A corner point of a feasible region is a point in the region which is the _________ of two boundary lines.

Concept understanding — Corner Point Theorem

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

The Intuition: Why Corners Win

The Precise Statement

Why This Matters for Exams

Method: Locating a corner point geometrically

The reasoning pattern

Q.A corner point of a feasible region is a point in the region which is the _________ of two boundary lines.

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