Corner points of the feasible region for an LPP are $(0, 2)$, $(3, 0)$, $(6, 0)$, $(6, 8)$ and $(0, 5)$. Let $F = 4x + 6y$ be the objective function. The Minimum value of $F$ occurs at (A) $(0, 2)$ only (B) $(3, 0)$ only (C) the mid point of the line segment joining the points $(0, 2)$ and $(3, 0)$ only (D) any point on the line segment joining the points $(0, 2)$ and $(3, 0)$

> [!TLDR] > The Corner Point Theorem says the optimum of a linear objective over a convex polygon occurs at a corner — but if two corners give the same value, every point on the edge joining them also gives that value. Here $F=12$ at both $(0,2)$ and $(3,0)$, so the minimum occurs along the entire segment between them. The correct option is **(D)**. The Corner Point Theorem is the backbone of solving any Linear Programming Problem (LPP) graphically. It states that if a feasible region is a convex polygon and the objective function is linear, then the maximum or minimum value occurs at one of the corner points. But there’s a subtle extension: if two adjacent corners give the *same* optimal value, then every point on the edge joining them also yields that same value. That’s exactly what happens here. Let’s work through it. 1. **List the corner points and compute $F = 4x + 6y$ at each.** - $(0, 2)$: $F = 4(0) + 6(2) = 12$ - $(3, 0)$: $F = 4(3) + 6(0) = 12$ - $(6, 0)$: $F = 4(6) + 6(0) = 24$ - $(6, 8)$: $F = 4(6) + 6(8) = 24 + 48 = 72$ - $(0, 5)$: $F = 4(0) + 6(5) = 30$ 2. **Identify the smallest value.** The smallest $F$ among these is $12$, which occurs at two corners: $(0,2)$ and $(3,0)$. 3. **Check if these two corners are adjacent on the feasible region.** The feasible region is a convex polygon whose vertices are given in order. The points $(0,2)$ and $(3,0)$ are connected by a straight edge — you can see this because no other corner lies between them along the boundary. They are adjacent. > [!WARNING] > A common mistake is to pick option (A) or (B), thinking the minimum occurs at *only one* corner. But when two adjacent corners give the same value, the entire edge is optimal — not just the endpoints. 4. **Apply the extension of the Corner Point Theorem.** …

Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let F = 4x + 6y

NCERT Exemplar · Q43

(0, 2)

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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 says the optimum of a linear objective over a convex polygon occurs at a corner — but if two corners give the same value, every point on the edge joining them also gives that value. Here $F=12$ at both $(0,2)$ and $(3,0)$ , so the minimum occurs along the entire segment between them. The correct option is (D).

The Corner Point Theorem is the backbone of solving any Linear Programming Problem (LPP) graphically. It states that if a feasible region is a convex polygon and the objective function is linear, then the maximum or minimum value occurs at one of the corner points. But there’s a subtle extension: if two adjacent corners give the same optimal value, then every point on the edge joining them also yields that same value. That’s exactly what happens here.

Let’s work through it.

List the corner points and compute $F = 4x + 6y$ at each.
- $(0, 2)$ : $F = 4(0) + 6(2) = 12$
- $(3, 0)$ : $F = 4(3) + 6(0) = 12$
- $(6, 0)$ : $F = 4(6) + 6(0) = 24$
- $(6, 8)$ : $F = 4(6) + 6(8) = 24 + 48 = 72$
- $(0, 5)$ : $F = 4(0) + 6(5) = 30$
Identify the smallest value.

The smallest $F$ among these is $12$ , which occurs at two corners: $(0,2)$ and $(3,0)$ .
Check if these two corners are adjacent on the feasible region.

The feasible region is a convex polygon whose vertices are given in order. The points $(0,2)$ and $(3,0)$ are connected by a straight edge — you can see this because no other corner lies between them along the boundary. They are adjacent.

Watch out

A common mistake is to pick option (A) or (B), thinking the minimum occurs at only one corner. But when two adjacent corners give the same value, the entire edge is optimal — not just the endpoints.

Apply the extension of the Corner Point Theorem. …

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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: Recognising multiple optimal solutions

Steps

Common Mistakes

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