A feasible region has corner points at (0, 4), (3, 2), and (6, 0). The objective function is P = 3x + 5y. A student argues that the optimal mix must be somewhere in the middle — not at an extreme corner — and evaluates P at the interior point (3, 2) only, concluding P = 19 is the maximum. What is the actual maximum value of P?
A20 — at corner point (0, 4)
B19 — the student's answer was correct
C18 — at corner point (6, 0)
DA higher value exists in the interior of the feasible region
P(0,4) = 0 + 20 = 20, P(3,2) = 9 + 10 = 19, P(6,0) = 18 + 0 = 18. The maximum is 20 at (0, 4). The Corner Point Theorem guarantees the optimum occurs at a vertex — never in the interior — because the objective function's level curves are straight lines that slide across the region until they exit at a corner. A point in the interior can always be improved by moving toward a boundary or corner.
Question 2 Multiple Choice
Which action correctly applies the Corner Point Theorem when solving a linear programming problem?
AEvaluate the objective function at many points inside the feasible region and find the largest
BEvaluate the objective function at each vertex of the feasible region and compare
CFind where the objective function equals zero and check nearby vertices
DEvaluate the objective function along each boundary edge and average the results
The Corner Point Theorem states that the maximum (or minimum) of a linear objective function over a closed polygonal feasible region always occurs at one of the vertices. The correct procedure is: (1) find all corner points by solving pairs of intersecting boundary lines, (2) evaluate the objective at each, (3) pick the best. Sampling interior points or averaging edges is both inefficient and unreliable.
Question 3 True / False
If a linear programming problem has a bounded feasible region, the Corner Point Theorem guarantees that the optimal value of the objective function occurs at one of the vertices.
TTrue
FFalse
Answer: True
True. A bounded feasible region is a closed polygon. Because the objective function is linear, its level curves are parallel lines. As you translate these lines in the direction of improving the objective, the last point of the feasible region touched before the line exits is always a corner vertex. This is why the theorem holds — and it only requires the region to be closed and the objective to be linear.
Question 4 True / False
A point in the interior of the feasible region can sometimes achieve a higher objective value than most corner points if the objective function has a steep slope.
TTrue
FFalse
Answer: False
False. No interior point can beat every corner point for a linear objective. The Corner Point Theorem is unconditional for bounded feasible regions — it applies regardless of the slope or direction of the objective function. Interior points lie on level curves strictly between two boundary values; moving toward the boundary always keeps the option to improve. If the objective function is parallel to a boundary edge, the entire edge is optimal, but each endpoint of that edge is still a corner.
Question 5 Short Answer
A student solves a linear programming problem by graphing the constraints, shading the feasible region, and then picking the point that 'looks like the best balance' near the middle of the region. Explain why this approach is structurally unreliable, and describe the correct procedure.
Think about your answer, then reveal below.
Model answer: The approach fails because the optimal value of a linear objective never lies in the interior of the feasible region — it always occurs at a corner vertex. 'Looking balanced' has nothing to do with maximizing a linear function. The correct procedure is: (1) graph all constraints to find the feasible region, (2) identify all corner points by solving the systems of equations formed at each intersection of boundary lines, (3) evaluate the objective function at every corner point, and (4) select the corner with the best value. This is the Corner Point Theorem and it is guaranteed by the geometry of linear functions over polygons.
The misconception stems from confusing optimization with 'balance.' A linear objective function values one unit of x and one unit of y at fixed rates regardless of where in the region you are. Moving toward a corner that weights the more valuable variable more heavily always improves the objective. Only when the objective is parallel to a boundary edge does an entire edge (including two corners) tie for optimal.