Page 864 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Solution

The constraints in this problem are identical to the constraints in Example 5, so the feasible region of this problem is also given by
Figure 11.3.3. The values of the objective function at the extreme points are given in the following table.

Extreme Point Value of

(0, 6) 36

(3, 6) 48

(9, 2) 48

(7, 0) 28

(0, 0) 0

We see that the objective function attains a maximum value of 48 at two adjacent extreme points, (3, 6) and (9, 2). This shows that
an optimal solution to a linear programming problem need not be unique. As we ask the reader to show in Exercise 10, if the
objective function has the same value at two adjacent extreme points, it has the same value at all points on the straight line
boundary segment connecting the two extreme points. Thus, in this example the maximum value of z is attained at all points on the
straight line segment connecting the extreme points (3, 6) and (9, 2).

EXAMPLE 7 The Feasible Region Is a Line Segment
Find values of and that minimize
subject to

Solution

In Figure 11.3.4 we have drawn the feasible region of this problem. Because one of the constraints is an equality constraint, the
feasible region is a straight line segment with two extreme points. The values of z at the two extreme points are given in the
following table.