Find the minimum and maximum values of the objective function subject to the given constraints?

Objective Function: C=4x+5y
Constraints: x>=0, y>=0, x+y<=6

My experience tells me to test the corners: (0,0); (0,6); (6,0)

C(0,0)=0
C(6,0)=24
C(0,6)=30

To find the minimum and maximum values of the objective function subject to the given constraints, we need to determine the feasible region and evaluate the objective function at the boundary points of the region.

Let's solve the constraints step by step:

1. x ≥ 0: This constraint means that the x-coordinate must be greater than or equal to zero. It represents the half-plane to the right of the y-axis.

2. y ≥ 0: This constraint means that the y-coordinate must be greater than or equal to zero. It represents the half-plane above the x-axis.

3. x + y ≤ 6: This constraint represents the region below the line x + y = 6. To find the boundary, we can set x + y = 6 and solve for y:
y = 6 - x

Now, let's plot these constraints on a graph:

The first two constraints create a quadrant in the first quadrant (above the x-axis and to the right of the y-axis).

The third constraint represents a line with an intercept of 6 on the y-axis and an intercept of 6 on the x-axis. The region below this line is the feasible region.

Next, we will evaluate the objective function at the vertices (corners) of the feasible region to find the minimum and maximum values:

1. Vertex A: (0, 0)
C = 4(0) + 5(0) = 0

2. Vertex B: (0, 6)
C = 4(0) + 5(6) = 30

3. Vertex C: (6, 0)
C = 4(6) + 5(0) = 24

By comparing the values of the objective function C at each vertex, we can find the minimum and maximum values:

Minimum value of C: 0 (occurs at vertex A)
Maximum value of C: 30 (occurs at vertex B)

To find the minimum and maximum values of the objective function subject to the given constraints, we can use the method of linear programming.

1. Graph the feasible region:
Start by graphing the constraints on a coordinate plane. Plot the points where each constraint intersects the axes and connect them to form the boundary lines of the feasible region. Shade the region that satisfies all the given constraints.

2. Identify the corner points:
The corner points are the vertices of the feasible region. These points need to be examined to determine the minimum and maximum values of the objective function.

3. Evaluate the objective function at each corner point:
Plug the x and y values of each corner point into the objective function, C = 4x + 5y, to find the corresponding objective function values.

4. Compare the objective function values:
Compare the values obtained from step 3 to identify the minimum and maximum values of the objective function.

Let's go through these steps:

1. Graph the feasible region:
Graph the constraints: x >= 0, y >= 0, and x + y <= 6.
Plot the points (0,0), (0,6), and (6,0), and connect them to form a triangle as the feasible region.

2. Identify the corner points:
The three corner points of the triangle are (0,0), (0,6), and (6,0).

3. Evaluate the objective function at each corner point:
Plug the x and y values of each corner point into the objective function C = 4x + 5y.

For (0,0), C = 4(0) + 5(0) = 0.
For (0,6), C = 4(0) + 5(6) = 30.
For (6,0), C = 4(6) + 5(0) = 24.

4. Compare the objective function values:
The minimum value of the objective function is 0, which occurs at the point (0,0).
The maximum value of the objective function is 30, which occurs at the point (0,6).

Therefore, the minimum value of the objective function C is 0, and the maximum value is 30, subject to the given constraints.