Final answer:
The statement that the minimum of an objective function occurs at a corner point of the feasible region is True. This reflects a fundamental principle in linear programming knowns as the corner point theorem or vertex method, where the optimal values are found at the vertices of the feasible region for a linear objective function over a convex polygon. Option A is the correct answer.
Step-by-step explanation:
The statement that if the minimum of an objective function exists, it will occur at one or more of the corner points of the feasible region is True. This is a fundamental concept in linear programming, a method used in mathematics to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. In linear programming, the feasible region is the set of all possible points that satisfy the constraints of the problem, typically represented by a polygon on a graph. The objective function is a formula that quantifies what you want to maximize or minimize, such as profit or cost.
According to the theory of linear programming, especially the fundamental theorem of linear programming, if the objective function is linear and the feasible region is a convex polygon, then the optimal value of the objective function occurs at one of the vertices, or corner points, of the feasible region. This happens because linear functions are flat, and their optimum values on a polygon will be found where the slope of the constraint boundary changes, which is at a vertex. The method of finding the optimal solution by checking the values at corner points is known as the corner point theorem or the vertex method.
Therefore, if the objective function has a minimum value, and we are working within the confines of linear programming, it will definitely be found at one or more of the corner points of the feasible region, assuming the feasible region has corners (i.e., it is bounded). In conclusion, option A. True is the correct answer.