Given the system of constraints, name all vertices of the feasible region. Then find the maximum value of the given objective fu

Response with clarification:

The constraints include:

• The first two constraints indicate that the solution needs to reside in the first quadrant, including the origin (

), as both x and y values are non-negative in this area

• Additionally, the third inequality corresponds to a solid line (due to the non-strict inequality) intersecting at (-9,0) and (0,3), with the shaded region extending towards the origin (verified by the zero-point test).

• The fourth inequality also represents a solid line (as the inequality is non-strict) that intersects at (5,0) and (0,5), and again the shaded area runs toward the origin (confirmed by the zero-point test)

Question 1)

The feasible region determined by these inequalities is:

                   Quadrilateral ABCD.

The vertices of the feasible region are:

A(0,3)

B(1.5,3.5)

C(5,0)

D(0,0)

Question 2)

The objective function is evaluated at the boundary points of the feasible region, which are the vertices.

The objective function is:

C=6x-4y

• Evaluating at A(0,3)

                C=6×0-4×3

                   Thus,

                 C= -12

• Evaluating at B(1.5,3.5)

              C=6×1.5-4×3.5

                    Thus,

                  C= -5

• Evaluating at C(5,0)

            The value of the objective function is:

                   C=6×5-4×0

                          Thus,

                        C=30

• [ TAG_89]]and at D(0,0)

                            C=6×0-4×0

                            Thus,

                          C=0

The highest value of the objective function is: 30

and it occurs at the vertex C(5,0)