Answer:
- Vertices of the feasible region: (5, 0), (1.5, 3.5), (0, 3), (0, 0)
- Maximum value of C: 30, at (x, y) = (5, 0).
Explanation:
1. Plot the constraints. (See the attached)
2. Find the region of overlap. This is the solution space, or "feasible region." (In the attached, this is the purple region where the red and blue regions overlap.)
3. Identify the corners of the feasible region. These are the points where the boundary lines of the feasible region intersect. (They are the vertices listed above.)
4. Evaluate the objective function at each of the corners of the feasible region to determine which maximizes the function. (See comment below.) For (x, y) = (5, 0), C = 6·5 -4·0 = 30.
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Comment on maximizing the objective function
This objective function has a positive coefficient (+6) for x and a negative coefficient (-4) for y. In order to maximize it, x must be made as large as possible at the same time y is made as small as possible. This means you will be seeking values of x and y that are as far as possible to the lower right of the feasible region. In this problem, the vertex (x, y) = (5, 0) matches that description. This analysis makes it unnecessary to evaluate C for other corners of the feasible region (unless you just want to).
For an objective function with both coefficients positive, I find it convenient to plot the objective function with C=0. This can help you find the vertex of the feasible region that will move the objective function line farthest from the origin.
Some problems are such that the variable values must be integers. In such cases it may take a little searching to find the precise set of integers that maximizes the objective function. That solution may or may not be near one of the vertices of the feasible region.