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Maxima and Minima
The theory of maxima and minima for functions of two variables is similar to the theory for one variable. The student should review the one-variable case at this time. DEFINITION Let z = f(x, y) be a function with domain D. f is said to have a maximum at a point (x0, y0) in D if f(x0,y0) ≥ f(x,y) for all (x, y) in D. The value f(x0, yo) is called the maximum value of f. A minimum and the minimum value of f are defined analogously. We shall first study functions defined on closed regions, which correspond to closed intervals. By a closed region in the plane we mean a set D defined by inequalities a ≤ x ≤ b, f(x) ≤ y ≤ g(x), where f and g are continuous and f(x) ≤ g(x) on [a, b]. D is called the region between f(x) and g(x) for a ≤ x ≤ b (Figure 11.7.1).
Figure 11.7.1 The points of D on the four curves x = a, x = b, y = f(x), y = g(x) are called boundary points. All other points of D are called interior points.
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Last Update: 2006-11-25