Single Digits by Chamberland Marc

Author:Chamberland, Marc
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2015-04-12T04:00:00+00:00

FIGURE 3.23: The surface and contours for f(x, y) = –(x2 – 1)(x2 – 2) – y2.

The function f(x, y) = –(x2 – 1)(x2 – 2) – y2 has two local maxima and one saddle point (figure 3.23). You could picture this as a landscape with two peaks and a mountain pass running between them. To travel from one side of a mountain range to the other side while staying as low as possible, you pass through the saddle point, the maximum along that path. On the other hand, to travel from one peak to the other while staying as high as possible, the saddle point is encountered again. This time the saddle point is the lowest point along the path.

Let’s get back to the question about three critical points. The proper question asks whether there can be a function with two local maxima but no other critical points. Surprisingly, such a function exists (figure 3.24). An example is

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