Find the minimum value of f(x, y) = y – x + 1 for the polygonal convex set determined by this system of inequalities.
x > 0 y > 0 2x + y< 4
the answer is -1. please tell me why
To find the minimum value of a function over a polygonal convex set determined by a system of inequalities, we can use linear programming techniques.
First, let's define the vertices of the polygonal convex set. From the given system of inequalities:
x > 0
y > 0
2x + y < 4
We can rewrite the third inequality as y < 4 - 2x. Now let's find the intersection points of the lines 2x + y = 4 and the coordinate axes:
When x = 0, y = 4.
When y = 0, 2x = 4, so x = 2.
So the vertices of the polygonal convex set are (0, 4), (2, 0), and the origin (0, 0).
Now, we need to evaluate the function f(x, y) = y - x + 1 at each of these vertices to find the minimum value.
For (0, 4): f(0, 4) = 4 - 0 + 1 = 5.
For (2, 0): f(2, 0) = 0 - 2 + 1 = -1.
For (0, 0): f(0, 0) = 0 - 0 + 1 = 1.
Among these values, the minimum value is -1, which occurs at the vertex (2, 0) of the polygonal convex set.
Therefore, the minimum value of f(x, y) = y - x + 1 for the given polygonal convex set is -1.