Final answer:
To find the minimum value of the function for the polygonal convex set determined by the given system of inequalities, find the vertices of the polygonal convex set and substitute the coordinates into the function.
Step-by-step explanation:
To find the minimum value of the function for the polygonal convex set determined by the given system of inequalities, you need to find the vertices of the polygonal convex set and substitute the coordinates into the function. The minimum value is the smallest value obtained. Let's solve the system of inequalities:
2x + 2y ≤ 16
-2x + y ≤ 24
-4x + 4y ≤ 2
To find the vertices, set each inequality to an equality and solve for x and y:
2x + 2y = 16 ⇒ x + y = 8 ⇒ y = 8 - x
-2x + y = 24 ⇒ y = 24 + 2x
-4x + 4y = 2 ⇒ x - y = -0.5 ⇒ y = x + 0.5
Solving the equations, we find the vertices (4, 4), (4, -4), and (-4, -4). Substituting the coordinates into the function f(x, y) = 5x + 4y, we get:
f(4, 4) = 5(4) + 4(4) = 40
f(4, -4) = 5(4) + 4(-4) = 28
f(-4, -4) = 5(-4) + 4(-4) = -36
The minimum value of the function is -36.