Question: In one study of the quantities x and y of natural gas that Western Europe should import from Norway and Siberia, respectively, it was assumed that the benefits were given by the function f (x, y) = 9x + 8y - 6(x + y)2. (The term -6(x + y)2 occurs because the world price of natural gas rises as total imports increase.) Because of capacity constraints, x and y must satisfy 0 ≤ x ≤ 5 and 0 ≤ y ≤ 3. Finally, for political reasons, it was felt that imports from Norway should not provide too small a fraction of total imports at the margin, so that x ≥ 2(y - 1), or equivalently -x + 2y ≤ 2. Thus, the optimization problem was cast as
max f (x, y) = 9x + 8y - 6(x + y)2 subject to 0 ≤ x ≤ 5, 0 ≤ y ≤ 3, -x + 2y ≤ 2
In the xy-plane, draw the set S of all points satisfying the three constraints, and then solve the problem.