To find the minimum and maximum values of the objective function [tex]\( F = 8x + 5y \)[/tex] over a set of vertices, we need to evaluate the function at each vertex and then identify the smallest and largest values obtained. The given vertices are [tex]\((0, 100)\)[/tex], [tex]\((0, 80)\)[/tex], [tex]\((80, 60)\)[/tex], [tex]\((80, 0)\)[/tex], and [tex]\((120, 0)\)[/tex].
1. Evaluate the objective function [tex]\(F(x, y) = 8x + 5y \)[/tex] at each vertex:
- At [tex]\((0, 100)\)[/tex]:
[tex]\[
F(0, 100) = 8(0) + 5(100) = 0 + 500 = 500
\][/tex]
- At [tex]\((0, 80)\)[/tex]:
[tex]\[
F(0, 80) = 8(0) + 5(80) = 0 + 400 = 400
\][/tex]
- At [tex]\((80, 60)\)[/tex]:
[tex]\[
F(80, 60) = 8(80) + 5(60) = 640 + 300 = 940
\][/tex]
- At [tex]\((80, 0)\)[/tex]:
[tex]\[
F(80, 0) = 8(80) + 5(0) = 640 + 0 = 640
\][/tex]
- At [tex]\((120, 0)\)[/tex]:
[tex]\[
F(120, 0) = 8(120) + 5(0) = 960 + 0 = 960
\][/tex]
2. The values of the objective function at the vertices are:
[tex]\[
F(0, 100) = 500, \quad F(0, 80) = 400, \quad F(80, 60) = 940, \quad F(80, 0) = 640, \quad F(120, 0) = 960
\][/tex]
3. The minimum value is the smallest of these values:
[tex]\[
\min(500, 400, 940, 640, 960) = 400
\][/tex]
4. The maximum value is the largest of these values:
[tex]\[
\max(500, 400, 940, 640, 960) = 960
\][/tex]
Therefore, the minimum value of the objective function [tex]\(F = 8x + 5y\)[/tex] is 400, and the maximum value is 960.
Minimum: [tex]\(400\)[/tex]
Maximum: [tex]\(960\)[/tex]
