find the max and min values of f(x,y)= 3x+y for the polygonal convex set determined by x ¡Ý 1, y ¡Ý 0, and x+ .5y ¡Ü 2

I don't understand your set symbols or terminology.

Compute 3x + y for each of the (x,y)points in the set and see for yourself where the maxima and minima are.

To find the maximum and minimum values of the function f(x, y) = 3x + y for the given polygonal convex set determined by x ≥ 1, y ≥ 0, and x + 0.5y ≤ 2, we can follow these steps:

Step 1: Plot the region determined by the inequalities.

- Draw the lines x = 1, y = 0, and x + 0.5y = 2 on a coordinate system.
- Shade the region above the x-axis and to the right of the line x = 1.
- Shade the region below the line x + 0.5y = 2.

Step 2: Determine the vertices of the polygonal convex set.

- The vertices of the polygonal convex set can be found at the intersection points of the lines and boundaries of the shaded region.

For x = 1 and y = 0:
- This point satisfies the inequalities, so it is a vertex.

For x + 0.5y = 2 and y = 0:
- Solving the equation x + 0.5y = 2 for x, we get x = 2.
- This point satisfies the inequalities, so it is a vertex.

For x + 0.5y = 2 and x = 1:
- Solving the equation x + 0.5y = 2 for y, we get y = 2.
- This point satisfies the inequalities, so it is a vertex.

So, the vertices of the polygonal convex set are (1, 0), (2, 0), and (1, 2).

Step 3: Substitute the vertices into the function f(x, y) = 3x + y and find the maximum and minimum values.

- Substitute each coordinate pair from the vertices into the function f(x, y) = 3x + y and calculate the corresponding values.

For (1, 0):
- f(1, 0) = 3(1) + 0 = 3

For (2, 0):
- f(2, 0) = 3(2) + 0 = 6

For (1, 2):
- f(1, 2) = 3(1) + 2 = 5

So, the maximum value of f(x, y) = 3x + y within the polygonal convex set is 6, and the minimum value is 3.