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Method For Finding Maxima and Minima on a Closed Region

If we know a function has a maximum or minimum, we can find it simply by finding the critical point. But usually we are not sure whether a function has a maximum or minimum. Here is a method that can be used when a function has a unique critical point in an open region. It is based on the fact that the Extreme Value Theorem holds for closed regions of the hyperreal plane as well as the real plane (because of the Transfer Principle).

Given a real open region D we can find a hyperreal closed region E which contains the same real points as D (Figure 11.7.11).

11_partial_differentiation-474.gif 11_partial_differentiation-475.gif
(a) (b)
Figure 11.7.11 Hyperreal Closed Regions

For example, if D is the real region

a < x < b, f(x) < y < g(x),

we can take for E the hyperreal region

a + ε ≤ x ≤ b - ε, f(x) + ε ≤ y ≤ g(x) - ε

where ε is positive infinitesimal.

If D is the whole real plane we can take for E the hyperreal region

-H ≤ x ≤ H, -H ≤ y ≤ H

where H is positive infinite.

METHOD FOR FINDING MAXIMA AND MINIMA ON AN OPEN REGION

When to Use z = f(x, y) is a smooth function whose domain is an open region D, and f has exactly one critical point.

Step 1

Set up the problem and sketch D if necessary.

Step 2

Compute ∂z/∂x and ∂z/∂y.

Step 3

Find the critical point (x0, y0) and the value f(x0,y0). If we already know there is a maximum (or minimum), it must be (x0, y0) and we can stop here.

Step 4

Find a hyperreal closed region E with the same real points as D.

Step 5

Compare f(x0,y0) with the values of f on the boundary of E.

CONCLUSION

f has a maximum at (x0, y0) if f(x0, y0) ≥ f(x, y) for every boundary point (x, y) of E. Otherwise f has no maximum.

A similar rule holds for the minimum.

Example 6
Example 7


Last Update: 2006-11-05