Final answer:
The maximum value of the objective function f(x,y)=7x+8y subject to given constraints is 23 at the point (1, 2), and the minimum value is 0 at the point (0, 0).
Step-by-step explanation:
To find the minimum and maximum values of the objective function f(x,y)=7x+8y, given the constraints x≥0, y≥0, 2x+y≤4, and x+4y≤8, we can use the method of linear programming. This involves plotting the constraints on a graph and identifying the feasible region. The maximum and minimum values of f(x,y) will occur at the vertices of the feasible region, and they can be found by evaluating the objective function at each vertex. Let's begin by plotting the constraints on a graph.
The points of intersection for the constraints give us the vertices of the feasible region (0, 0), (2, 0), (1, 2), and (0, 1). Evaluating the objective function f(x,y) at these vertices, we get:
- f(0, 0) = 7(0) + 8(0) = 0
- f(2, 0) = 7(2) + 8(0) = 14
- f(1, 2) = 7(1) + 8(2) = 23
- f(0, 1) = 7(0) + 8(1) = 8
The maximum value is 23 at the point (1, 2), and the minimum value is 0 at the point (0, 0). The objective function reaches its minimum and maximum values at the corners of the feasible region, which is a common characteristic in linear programming problems.