What are extreme points in the context of linear programming?

Extreme points, in the context of linear programming, refer specifically to the feasible solution points that exist at the vertices of the feasible region. The feasible region is defined by the constraints of the linear programming problem, and it is typically represented geometrically as a polygon or polyhedron. The extreme points are particularly significant because, according to the fundamental theorem of linear programming, the optimal solution to a linear programming problem will always occur at one of these extreme points.

This is due to the nature of linear objective functions, which form straight lines or planes. As a result, when optimizing (maximizing or minimizing) the objective function, the most effective values will be found at the vertices where the boundaries defined by the constraints meet. Identifying these extreme points is essential for efficiently solving linear programming problems, as they provide key candidates for the optimal solution.

The other options do not accurately describe extreme points in linear programming. For instance, points where the objective function intersects the feasible region may not always be vertices, and infeasible solutions do not exist within the feasible region at all. Random points within the feasible region could be valid solutions, but they may not possess the optimality property found at extreme points.