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 shaded region from the graph. The points are (3, 4), (4, 8), (7, 7), and (9, 5).
2. Write down the objective function given: \(z = 5x + 6y\).
3. Calculate the value of the objective function at each corner point by substituting the coordinates into the function:
4. For each point \((x, y)\), compute \(z = 5 \times x + 6 \times y\).
5. Compare the calculated values of \(z\) at all corner points to determine which is the maximum and which is the minimum.

Key Concepts

Objective Function

An objective function is a linear expression that you want to maximize or minimize, such as z = 5x + 6y. It assigns a value to each point (x, y) in the feasible region, helping to determine the best solution based on given criteria.

Feasible Region and Corner Points

The feasible region is the set of all points that satisfy the system of inequalities, often shown as a shaded polygon. The corner points (vertices) of this region are critical because the maximum or minimum values of a linear objective function occur at these points.

Evaluating the Objective Function at Corner Points

To find the maximum or minimum value of the objective function, substitute the coordinates of each corner point into the function. Comparing these values identifies which corner yields the highest or lowest result, solving the optimization problem.