Name all the values of x that are not in the domain of f(x)=2-x^2/x+5.

Answer: x= -5

2)Find the minimum value of f(x,y)=2x-y+2 for the polygonal convex set determined by this system of inequalities: x >/= 1, x </= 3, y</=0, 1/2x+y</=5

Answer: -0.5

To find the values of x that are not in the domain of the function f(x) = (2 - x^2)/(x + 5), we need to identify the values that would make the denominator equal to zero.

The domain of a rational function excludes any value of x that would result in a division by zero. In this case, we set the denominator, x + 5, equal to zero and solve for x:

x + 5 = 0
x = -5

Therefore, the value x = -5 is not in the domain of the function f(x).

Moving on to the second question, let's find the minimum value of the function f(x, y) = 2x - y + 2 within the polygonal convex set determined by the given system of inequalities:

x ≥ 1,
x ≤ 3,
y ≤ 0,
(1/2)x + y ≤ 5

A polygonal convex set is a region in a coordinate plane that is both convex and bounded. In this case, we have a region determined by the system of inequalities.

To find the minimum value of a function within this region, we need to identify the critical points and compare the values of the function at those points to determine the minimum.

First, we find the vertices of the polygonal convex set by solving the system of inequalities. The vertices are the intersection points of the boundary lines:

1. (x = 1, y = 0)
2. (x = 3, y = 0)
3. (x = 2, y = 0)
4. Solve the equation (1/2)x + y = 5 for x = 1:
(1/2)(1) + y = 5
1/2 + y = 5
y = 4.5 (approx.)
So, we have another vertex (x = 1, y = 4.5)

Now, we evaluate the function f(x, y) = 2x - y + 2 at each of these vertices:

1. f(1, 0) = 2(1) - 0 + 2 = 4
2. f(3, 0) = 2(3) - 0 + 2 = 8
3. f(2, 0) = 2(2) - 0 + 2 = 6
4. f(1, 4.5) = 2(1) - 4.5 + 2 = 0.5

Comparing these values, we find that the minimum value of f(x, y) within the polygonal convex set is -0.5 at (x = 1, y = 4.5).