Final answer:
The maximum value of the objective function in a linear programming problem is often found at the vertices of the feasible region, not the midpoints between vertices. The correct answer is D. No, the maximum value may not occur at the midpoints.
Step-by-step explanation:
In the context of linear programming, when the objective function reaches its maximum value at adjacent vertices of a feasible region, this does not guarantee that the function will also reach maximum values at the midpoints or any other points between the vertices. The maximum value of the objective function in a linear programming problem occurs at the vertices of the feasible region. This is a result of the linear nature of both the objective function and constraints in such problems. Therefore, having maximum values at the vertices (2,5) and (7,3) does not imply that points (2.5, 4.8) and (6.5, 3.2) also yield maximum values; the latter are simply points along the edge connecting the vertices. In fact, linear programming problems often exhibit the property that the optimal solution is found at a vertex, or can extend along an edge if there is multiple optimal solutions forming a line segment, but is not generally found at arbitrary points between vertices.
The correct answer to the student's question is D. No, the maximum value may not occur at the midpoints. This is because within the feasible region defined by linear constraints, the objective function's value changes linearly, and the maximum value, due to the properties of linear functions, is found only at the corners (vertices) of the feasible region or along an entire edge if multiple vertices have the same value for the objective function.