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Home Partial Differentiation Maxima and Minima Method For Finding Maxima and Minima on a Closed Region
 
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Method For Finding Maxima and Minima on a Closed Region

When to Use z = f(x, y) is continuous on a closed region D and smooth on the interior of D.

Step 1

Set the problem up and sketch D.

Step 2

Compute ∂z/∂x and ∂z/∂y.

Step 3

Find the critical points of f, if any, and the value of f at each critical point.

Step 4

Find the maximum and minimum of f on the boundary of D. This can be done by solving for z as a function of x or y alone and using the method for one variable.

CONCLUSION

The largest of the values from Steps 3 and 4 is the maximum value, and the smallest is the minimum value.

It is convenient to record the results of Steps 3 and 4 on the sketch of D.

Example 1: Closed Rectangle

In many problems we are to maximize a function of three variables which are related by a side condition. We wish to find the maximum or minimum of

w = F(x, y, z) given the side condition

g(x, y, z) = 0.

To work a problem of this type we use the side condition to get w as a function of just two independent variables and then proceed as before.

Example 2: Maximum Volume

Home Partial Differentiation Maxima and Minima Method For Finding Maxima and Minima on a Closed Region

Last Update: 2006-11-25