The Fundamental Theorem of Linear Programming

A weak version of what is sometimes called the fundamental theorem of linear programming states that the extremal values of a linear function over a convex polygonal region are attained at corners of the region. (Moreover, if an extremum is attained at two corners then it is attained everywhere on the line segment connecting them.) The function is called the objective function, and often represents profit or some other quantity to be optimized. The region is called the feasible set and represents all options for the variables and that satisfy the constraints in the problem. Points on the red line in the Demonstration are those points in the feasible set that produce the given value of the objective function.