Maximize and minimize p = x + 2y subject to
x + y ≥ 8
x + y ≤ 10
x − y ≤ 8
x − y ≥
−8.
Minimum:
p = 1
x = 2
y = 3
Maximum:
p = 4
x = 5
y = 6
To solve this problem, we need to find the maximum and minimum values of p (the objective function) subject to the given constraints. The constraints in this case are four inequalities.
One approach to solve this type of problem is by using the method of linear programming. Here are the steps:
1. Graph the feasible region: Start by graphing each constraint on a coordinate plane. Shade the region that satisfies all the constraints.
The feasible region in this case is the area that satisfies all the given inequalities.
2. Identify the corner points: On the graph, locate the corner points of the feasible region. These are the points where the boundaries of the constraints intersect.
3. Calculate the objective function for each corner point: Plug the coordinates of each corner point into the objective function p = x + 2y and calculate the resulting value.
4. Determine the maximum and minimum: Compare the values of p for each corner point. The maximum value of p represents the largest possible solution, while the minimum value represents the smallest possible solution.
From the given example, we have the following values:
Minimum:
p = 1
x = 2
y = 3
Maximum:
p = 4
x = 5
y = 6
These values represent the minimum and maximum solutions, respectively.
Note: It's important to note that this is just an example solution. To get the precise values, you will need to follow the steps explained above or use software that can solve linear programming problems.