Final answer:
True, an optimal solution to a linear programming problem is often found at a corner point or vertex of its feasible region, as per the Fundamental Theorem of Linear Programming, as long as the problem is bounded.
Step-by-step explanation:
The statement that an optimal solution to a linear programming problem can be found at a corner point of its feasible region, assuming the problem is not unbounded, is true. In linear programming, this is due to the Fundamental Theorem of Linear Programming which states that if there is an optimal solution, it will occur at a vertex, or corner point, of the feasible region. The feasible region is defined by a set of linear inequalities that represent constraints of the problem. Because these constraints are linear, the region they enclose is a convex polyhedron, and the objective function being optimized (maximized or minimized) is also a linear function.
To visualize, consider a two-dimensional case where the feasible region is a polygon on a graph. Each corner point represents a potential solution where two or more constraints intersect. As we move along the edges of this polygon, the value of the objective function changes linearly. When optimizing, the highest or lowest value of the objective function (depending on whether we are maximizing or minimizing) will be found at one of these intersections, hence at a corner point. This principle holds in higher dimensions as well, with the feasible region being a polyhedral set and corner points being the intersection of hyperplanes that define the constraints.