Example 4: No Maximum

Show that the function z = x² + 2y² has no maximum. The domain is the whole plane. We have

There is one critical point at (0, 0). At this point, z = 0. This is not a maximum because, for example, z = 3 at (1, 1). Hence z has no maximum.

Figure 11.7.9

Notice that z has a minimum at (0,0) because x² + 2y² is always ≥ 0 (Figure 11.7.9).