Corner Point Calculator

A corner point (also called a vertex) is a point where two or more boundary lines of the feasible region intersect. According to the Corner Point Theorem, the optimal value of a linear objective function over a bounded feasible region always occurs at one of these corner points, so you only need to evaluate the function at the vertices.

First, identify all corner points of the feasible region by solving each pair of boundary equations simultaneously. Then substitute each corner point (x, y) into your objective function P = cx·x + cy·y. The corner point that produces the largest value is the maximum; the one producing the smallest value is the minimum.

The point that maximizes the objective function is always one of the corner points of the feasible region (assuming the region is bounded). This calculator evaluates P at every corner point and highlights the one where P reaches its greatest value.

No. A feasible region defined by a system of linear inequalities is always convex — meaning it forms a single, connected region with no gaps or holes. This convexity property is what guarantees the optimal solution lies at a vertex.

Not necessarily. If the constraints are contradictory — for example, requiring x + y ≤ 2 and x + y ≥ 5 simultaneously — then no point satisfies all constraints at once and the feasible region is empty (infeasible). This calculator will report zero corner points in that scenario.

To find corner points algebraically, take each pair of constraint boundary lines (written as equalities) and solve the 2×2 system simultaneously. Then check whether the resulting (x, y) point satisfies all other constraints including x ≥ 0 and y ≥ 0. Points that satisfy all constraints are valid corner points.

This calculator assumes all constraints are of the form a·x + b·y ≤ k, and it automatically includes the non-negativity constraints x ≥ 0 and y ≥ 0. All boundary intersections — including with the axes — are tested and filtered to keep only feasible vertices.

Yes. If the objective function is parallel to one of the constraint boundary edges, then all points along that edge (including two corner points) yield the same optimal value. In this case the problem has infinitely many optimal solutions, though this calculator will still report all corner point values correctly.