Find the value of the objective function z = 2x + 3y at each corner of the graphed region shown. What is the maximum value of the objective function? What is the minimum value of the objective function?

Verified step by step guidance

1. Identify the corner points of the graphed region: (0, 0), (0, 13), (4, 5), and (5, 0).
2. Substitute each corner point into the objective function z = 2x + 3y to find the value of z at each point.
3. For the point (0, 0), calculate z = 2(0) + 3(0).
4. For the point (0, 13), calculate z = 2(0) + 3(13).
5. For the point (4, 5), calculate z = 2(4) + 3(5).

Key Concepts

Objective Function

An objective function is a mathematical expression that defines the goal of an optimization problem, typically in the form of maximizing or minimizing a value. In this case, the objective function is z = 2x + 3y, which needs to be evaluated at specific points to find its maximum and minimum values within a defined region.

Corner Points

Corner points, or vertices, of a feasible region in linear programming are the points where the boundary lines intersect. These points are critical because, according to the Fundamental Theorem of Linear Programming, the maximum and minimum values of the objective function will occur at one of these corner points within the feasible region.

Feasible Region

The feasible region is the area on a graph that satisfies all the constraints of a linear programming problem. It is typically bounded by the lines representing the constraints, and the objective function is evaluated at the corner points of this region to determine the optimal solution.