Find the maximum and minimum values of the function for the polygonal convex set determined by the given system of inequalities.
x + y > 2
8x – 2y < 16
4y < 6x + 8
f(x, y) = 4x + 7y
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maximize 4x+7y given x+y>2, 8x-2y<16, 4y<6x+8
same for minimize
To find the maximum and minimum values of f(x, y) for the polygonal convex set determined by the system of inequalities, we need to find the vertices of the feasible region first.
Step 1: Graph the inequalities
Let's graph the three inequalities to determine the feasible region.
The first inequality, x + y > 2, represents the area above the line x + y = 2.
The second inequality, 8x - 2y < 16, represents the area below the line 8x - 2y = 16.
The third inequality, 4y < 6x + 8, represents the area below the line 4y = 6x + 8.
Step 2: Determine the intersection points
The feasible region is the area where all three inequalities overlap. Find the points where the lines intersect. These intersection points will be the vertices of the polygonal convex set.
Step 3: Evaluate the function at each vertex
Plug the x and y coordinates of each vertex into the function f(x, y) = 4x + 7y to find the corresponding values of f(x, y).
Step 4: Determine the maximum and minimum values
Among the values obtained in step 3, the largest value will be the maximum value of the function, and the smallest value will be the minimum value.
Note: It is not mentioned whether the inequalities are strict or non-strict (greater than or equal to, less than or equal to). If the inequalities are strict, i.e., greater than (>) and less than (<), you will need to consider the open regions on each line as well. That means the maximum and minimum may not occur at the vertices.
Following these steps, you can find the maximum and minimum values of the function for the given polygonal convex set determined by the system of inequalities.