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=40x+50y

Verified step by step guidance

1. Identify the corner points of the feasible region from the graph. The points are (0, 0), (0, 10), (6, 8), and (10, 0).
2. Write down the objective function: \(z = 40x + 50y\).
3. Calculate the value of the objective function at each corner point by substituting the coordinates into the function:
4. At (0, 0): \(z = 40(0) + 50(0)\)
5. At (0, 10): \(z = 40(0) + 50(10)\)
6. At (6, 8): \(z = 40(6) + 50(8)\)
7. At (10, 0): \(z = 40(10) + 50(0)\)
8. Compare the calculated values to determine which is the maximum and which is the minimum value of the objective function.

Key Concepts

Objective Function

An objective function is a mathematical expression that defines the goal of an optimization problem, often to maximize or minimize a value. In this problem, z = 40x + 50y represents the objective function, where x and y are variables, and the goal is to find the maximum and minimum values of z within the feasible region.

Feasible Region and Corner Points

The feasible region is the set of all possible points (x, y) that satisfy the problem's constraints, shown as the shaded area on the graph. The corner points (vertices) of this region are critical because, according to the linear programming theory, the maximum and minimum values of the objective function occur at these points.

Evaluating the Objective Function at Corner Points

To find the maximum and minimum values of the objective function, substitute the coordinates of each corner point into the function z = 40x + 50y. Calculate z for each vertex, then compare these values to identify which is the largest (maximum) and which is the smallest (minimum).