I have two problemson polygonal convex sets which I do not understand.

The first: Find the minimum value of f(x,y) = 2x-y+2 for the polygonal convex set determined by this system of inequalities:

x is more than or equal to 1
x is less than or equal to 3
y is more than or equal to 0
1/2x + y is less than or equal to 5

The second problem:

This system forms a polygonal convex set: x is less than or equal to 0;
y is more than or equal to -x, if x is between 0 and -6.

y is less than or equal to 10; 2x + 3y is more than or equal to 6, if x is between -12 and -6.

What is the area of the closed figure?

I do not have a clue as to how to do either of these problems.

I don't know what a 'polygonal' or a 'convex' set means. Your four inequalities define a trapezoidal region of x,y space. You can easily plot it on a graph. The lowest value of 2x - y + 2 in that space occurs where x is least (1) and y is highest (4.5). At that corner of the trapezoid,

2x-y+2 = 2 - 9 + 2 = -5

If that is what is meant by your problem, you should be able to answer the second question by a similar procedure

To solve both of these problems, we need to understand how to work with polygonal convex sets and find minimum values of functions over these sets, as well as calculate the area of a closed figure.

Problem 1: Finding the minimum value of a function over a polygonal convex set.

Step 1: Find the vertices of the polygonal convex set:
The system of inequalities defines a polygonal convex set. To find the vertices, we need to solve the equality conditions for each inequality.

From the first inequality:
x >= 1, we get the point (1, 0) since y can take any value.

From the second inequality:
x <= 3, we get the point (3, 0) since y can take any value.

From the third inequality:
y >= 0, we get the points (1, 0) and (3, 0) since x can take any value.

From the fourth inequality:
(1/2)x + y <= 5, we get the points (1, 0) and (3, 0) again, since x can take any value.

So, the polygonal convex set is a line segment between the points (1, 0) and (3, 0).

Step 2: Calculate the value of the function at each vertex:
Substitute the x and y coordinates of each vertex into the function f(x, y) = 2x - y + 2.

For the vertex (1, 0):
f(1, 0) = 2(1) - 0 + 2 = 4.

For the vertex (3, 0):
f(3, 0) = 2(3) - 0 + 2 = 8.

Step 3: Find the minimum value:
Compare the values obtained at each vertex to find the minimum value. In this case, the minimum value is 4 at the vertex (1, 0).

Therefore, the minimum value of f(x, y) = 2x - y + 2 over the given polygonal convex set is 4.

Problem 2: Finding the area of a closed figure defined by a polygonal convex set.

Step 1: Find the vertices of the polygonal convex set:
The given system of inequalities defines a polygonal convex set. To find the vertices, we need to solve the equality conditions for each inequality in different intervals of x.

From the inequality x <= 0:
We get the point (0, 0) since y can take any value.

From the inequality y >= -x:
We get the line y = -x.

From the inequality y <= 10:
We get the point (0, 10) since x can take any value.

From the inequality 2x + 3y >= 6:
We get the line 2x + 3y = 6.

Now, we need to find the intersection points of the lines y = -x and 2x + 3y = 6.

Solving these two equations simultaneously, we find the intersection point (-3, 3).

So, the polygonal convex set is a triangle formed by the vertices (0, 0), (0, 10), and (-3, 3).

Step 2: Calculate the area of the triangle:
To find the area of the triangle, we can use the formula for the area of a triangle given its vertices.

Let A, B, and C be the vertices of the triangle, with coordinates (x1, y1), (x2, y2), and (x3, y3), respectively.

The area of the triangle ABC = (1/2) * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.

Using the coordinates of the vertices, we can calculate the area of the triangle as follows:

Area = (1/2) * |0(10 - 3) + 0(3 - 10) + (-3)(0 - 3)|
Area = (1/2) * |-9|
Area = 4.5

Therefore, the area of the closed figure defined by the given polygonal convex set is 4.5 square units.