Final answer:
To find the minimum value of the given function, we solve the pairs of inequalities to find the vertices of the convex set and then substitute them into the function. The vertex that gives the minimum function value is the answer. In this case, it's the point (-3, 4), corresponding to option b.
Step-by-step explanation:
To find the minimum value of the function f(x, y) = 2x + y for the convex set determined by the given system of inequalities, we first need to identify the vertices of the feasible region. To do so, we can solve the system of equations formed by taking two inequalities at a time.
- 3x + 4y = 19 and -3x + 7y = 25
- -3x + 7y = 25 and -6x + 3y = -27
- -6x + 3y = -27 and 3x + 4y = 19
Once we have the vertices, we can substitute them into the function f(x, y) and find the values. The option that gives the minimum value is the correct answer. Let's solve each pair of equations:
- Solving equations 1 and 2 gives the intersection point (-3, 7).
- Solving equations 2 and 3 gives the intersection point (-3, 4).
- Solving equations 3 and 1 gives the intersection point (5, 1).
The function values at these points are as follows:
- f(-3, 7) = 2(-3) + 7 = 1
- f(-3, 4) = 2(-3) + 4 = -2
- f(5, 1) = 2(5) + 1 = 11
Based on these calculations, the minimum value of the function is at the point (-3, 4), which corresponds to option b.