In Exercises 1–4, find the value of the objective function at each corner of the graphed region. What is the maximum value of the objective function? What is the minimum value of the objective function? 1. Objective Function z=5x+6y

Verified step by step guidance

1. Identify the corner points of the graphed region: (3, 4), (4, 8), (7, 7), and (9, 5).
2. Substitute each corner point into the objective function z = 5x + 6y to find the value of z at each point.
3. Calculate z for the point (3, 4): z = 5(3) + 6(4).
4. Calculate z for the point (4, 8): z = 5(4) + 6(8).
5. Calculate z for the point (7, 7): z = 5(7) + 6(7).

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 given as z = 5x + 6y, where x and y are variables representing quantities to be optimized. The values of the objective function are evaluated at specific points, often referred to as corner points in a feasible region.

Feasible Region

The feasible region is the set of all possible points that satisfy a given set of constraints in a linear programming problem. It is typically represented graphically as a shaded area on a coordinate plane. The corner points of this region are critical for optimization, as they are where the maximum and minimum values of the objective function are likely to occur.

Corner Points

Corner points, or vertices, of the feasible region are the intersection points of the constraint lines in a linear programming problem. These points are essential for finding the optimal solution, as the maximum and minimum values of the objective function will occur at one of these corner points. In the provided graph, the corner points are labeled and represent the potential solutions to evaluate for the objective function.