Figure 16 shows the feasible set for the nutrition problem discussed in Example 1 of Section 3.3, along with the purple line of final cost passing through the optimal point. The figure shows that, as long as the slope of the final cost line is between -3 and - 23, the optimal solution will still be at the same vertex.

(a) Suppose that the cost of a cup of rice is changed from 21¢ to A¢ but that the cost of a cup of soybeans remains at 14¢. Show that the lines of constant cost will each have slope - A/14.

(b) Show that, when - 3 ≤ - A/14 ≤ - 23, then 28/3 ≤ A ≤ 42. Conclude that the interval [28/3, 42] is the range of optimality for the cost of a cup of rice.

(c) Use similar reasoning to show that the range of optimality for the cost of a cup of soybeans is [7, 31.5].

calories protein m = -3 lealen