To solve this optimization problem, we need to follow the procedure for linear programming:

1. Identify the feasible region: This region is typically defined by a set of inequalities, and the vertices (corner points) of this region are where we check the objective function.

2. Objective function: You have two objective functions:

   • $z = 5x + 4y$ for part (a)
   • $z = x + 2y$ for part (b)

3. Step-by-step process:

   • Find the coordinates of the vertices of the feasible region.
   • Plug the coordinates of each vertex into the objective function to calculate the value of $z$.
   • Determine the maximum and minimum values of $z$ and the corresponding points.

Since you have provided an objective function but not the specific inequalities that form the feasible region or the graph itself, I'll guide you on how to approach it:

Part 1 (a)

For $z = 5x + 4y$:

• Maximization:

  1. Find the values of $z$ at each vertex by substituting $(x, y)$ coordinates into the equation $z = 5x + 4y$.
  2. The maximum value of $z$ will be the highest among these values.

• Minimization:

  1. Similarly, substitute the $(x, y)$ coordinates into $z = 5x + 4y$ to find the minimum value.

Once you have the coordinates of the vertices, you can compute the objective function at those points.

Part 2 (b)

For $z = x + 2y$:

• Follow the same steps as above, substituting the vertices of the feasible region into $z = x + 2y$, and then find both the maximum and minimum values.

Without the graph or the list of inequalities defining the feasible region, we can't provide the exact values yet. However, here is the general process to solve it.

If you want, you can upload the graph or provide the inequalities, and I can guide you through solving it step by step!

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Would you like me to help with specific values, or do you have any questions about the steps?

Related Questions:

1. What are the coordinates of the vertices in the feasible region?
2. How do we find the feasible region from inequalities?
3. Can the objective function have multiple maximum or minimum points?
4. How does linear programming ensure that the maximum and minimum occur at the vertices?
5. What happens if the feasible region is unbounded?

Tip:

Always check the vertices of the feasible region when solving linear programming problems since these points determine the maximum and minimum values of the objective function.