The objective function is \( z=6 x+4 y \). A. Find the value of the objective function at each corner of the graphed region. B. Find the maximum value of the objective function. C. Find the minimum value of the objective function.

To solve the problem, we need to follow these steps: ### A. Find the value of the objective function at each corner of the graphed region. 1. **Identify the corner points** of the feasible region. These points are typically found by solving the system of inequalities that define the region. The corner points are where the constraints intersect. 2. **Evaluate the objective function** \( z = 6x + 4y \) at each corner point. Assuming you have the corner points from the graph (let's denote them as \( (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) \)), you would calculate: - \( z_1 = 6x_1 + 4y_1 \) - \( z_2 = 6x_2 + 4y_2 \) - \( z_3 = 6x_3 + 4y_3 \) - \( z_4 = 6x_4 + 4y_4 \) ### B. Find the maximum value of the objective function. After calculating the values of \( z \) at each corner point, you would compare these values to find the maximum: \[ \text{Maximum value} = \max(z_1, z_2, z_3, z_4) \] ### C. Find the minimum value of the objective function. Similarly, to find the minimum value, you would look for the smallest value among the calculated \( z \) values: \[ \text{Minimum value} = \min(z_1, z_2, z_3, z_4) \] ### Example Let's say the corner points of the feasible region are \( (0, 0), (0, 4), (2, 3), (3, 0) \). 1. Calculate \( z \) at each corner: - At \( (0, 0) \): \( z = 6(0) + 4(0) = 0 \) - At \( (0, 4) \): \( z = 6(0) + 4(4) = 16 \) - At \( (2, 3) \): \( z = 6(2) + 4(3) = 12 + 12 = 24 \) - At \( (3, 0) \): \( z = 6(3) + 4(0) = 18 \) 2. Values of \( z \): - \( z(0, 0) = 0 \) - \( z(0, 4) = 16 \) - \( z(2, 3) = 24 \) - \( z(3, 0) = 18 \) 3. Maximum value: - \( \max(0, 16, 24, 18) = 24 \) 4. Minimum value: - \( \min(0, 16, 24, 18) = 0 \) ### Conclusion - **Maximum value of the objective function**: 24 - **Minimum value of the objective function**: 0 Please provide the specific corner points or constraints if you need a more tailored solution!