Objective function and constraints make up a linear optimization problem.
The values of the objective function are 4, 16 and 40.
Given
[tex]z = 2x + 2y[/tex]
Subject to
[tex]x \le 6[/tex]
[tex]y \ge 2[/tex]
[tex]2x - y \le -2[/tex]
First, we plot the graphs of the constraints.
See attachment for the graphs of [tex]x \le 6[/tex], [tex]y \ge 2[/tex] and [tex]2x - y \le -2[/tex]
From the attached graph, the corner points are:
[tex](x_1,y_1) = (0,2)[/tex]
[tex](x_2,y_2) = (6,2)[/tex]
[tex](x_3,y_3) = (6,14)[/tex]
So, the value of the objective function at these points are calculated as follows:
[tex]z = 2x + 2y[/tex]
[tex]z = 2 \times 0 + 2 \times 2 = 4[/tex]
[tex]z = 2 \times 6 + 2 \times 2 = 16[/tex]
[tex]z = 2 \times 6 + 2 \times 14 = 40[/tex]
So, the values are: 4, 16 and 40
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