Find the maximum and minimum values of f(x,y)=3x+y for the polygonal convex set determined by x>or equal to 1, y>or equal to 0, and x+0.5y<or equal to 2.

I worked it out and got Maximum: 6; and Minimum: 3.

Is that right?

To find the maximum and minimum values of the function f(x, y) = 3x + y for the given polygonal convex set, you can follow these steps:

Step 1: Determine the vertices of the polygonal convex set.
The given conditions x ≥ 1, y ≥ 0, and x + 0.5y ≤ 2 represent inequalities that define the polygonal convex set. By graphing these inequalities, you can identify the vertices of the set.

- The inequality x ≥ 1 represents a vertical line passing through x = 1.
- The inequality y ≥ 0 represents the non-negative y-axis.
- The inequality x + 0.5y ≤ 2 represents a diagonal line with a slope of -0.5 passing through the point (2, 0) and (0, 4).

By examining the intersection points of these lines, you can determine the vertices of the polygonal convex set.

Step 2: Evaluate the function at each vertex.
Calculate the value of the function f(x, y) = 3x + y at each vertex of the polygonal convex set. This will give you a set of values that can help find the maximum and minimum.

Step 3: Compare the values and determine the maximum and minimum.
Once you have the values of the function at each vertex, compare them to find the maximum and minimum values.

Now, let's go through the steps:

Step 1: By graphing the given inequalities, we can determine the vertices of the polygonal convex set. The vertices are:
A(1, 0)
B(1, 2)
C(0, 4)
D(2, 0)

Step 2: Evaluate the function at each vertex:
f(A) = 3(1) + 0 = 3
f(B) = 3(1) + 2 = 5
f(C) = 3(0) + 4 = 4
f(D) = 3(2) + 0 = 6

Step 3: Compare the values:
The maximum value is 6 (obtained at vertex D), and the minimum value is 3 (obtained at vertex A). Therefore, your answer of Maximum: 6 and Minimum: 3 is correct.