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 shaded region from the graph. The corner points are (0, 0), (0, 13), (4, 5), and (5, 0).
2. Substitute each corner point into the objective function z = 2x + 3y to calculate the value of z at each point. For example, for the point (0, 0), substitute x = 0 and y = 0 into the equation.
3. Repeat the substitution for the other corner points: (0, 13), (4, 5), and (5, 0). For each point, calculate the value of z = 2x + 3y.
4. Compare the calculated values of z for all the corner points. The maximum value of z corresponds to the maximum value of the objective function, and the minimum value of z corresponds to the minimum value of the objective function.
5. State the corner point where the maximum value occurs and the corner point where the minimum value occurs, along with their respective z values.

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 are the points where the boundary lines intersect. In linear programming, the maximum and minimum values of the objective function occur at these corner points. The graph shows the corner points (0,0), (0,13), (4,5), and (5,0), which are essential for evaluating the objective function.

Feasible Region

The feasible region is the area on a graph that satisfies all constraints of a linear programming problem. It is typically bounded by the lines representing the constraints. The shaded region in the graph indicates the feasible region where the objective function will be evaluated to determine its maximum and minimum values.