Extreme Values Of A Function Of Two Variables

For a function of one choice variable, an extreme value is represented graphically by the peak of a hill or the bottom of a valley in a two-dimensional graph. With two choice variables, the graph of the function—z = f(x, y)—becomes a surface in a 3-space, and while the extreme values are still to be associated with peaks and bottoms, these "hills" and "valleys" themselves now take on a three-dimensional character. They will, in this new context, be shaped like domes and bowls, respectively. The two diagrams in Fig. 11.2 serve to illustrate. Point a in diagram a, the peak of a dome, constitutes a maximum; the value of z at this point is larger than at any other point in its immediate neighborhood. Similarly, point b in diagram b, the bottom of a bowl, represents a minimum; everywhere in its immediate neighborhood the value of the function exceeds that at point b.

Figure 11.2

First-Order Condition

For the function

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