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Unformatted text preview: Section 15.8 Lagrange Multipliers Finding minimum and maximum values for functions subject to certain constraints Recall that in the last section, to determine the absolute maximum and minimum values of a function, we needed to consider the values of that function on the boundary. Such a problem is equivalent to determining the minimum and maximum values of a function subject to some particular constraint. In this section, we shall consider this problem in more detail. We outline through an example. Consider the following situation which is typical in a business situation: A firm manufactures a commodity at two different factories. The total cost of manufacturing depends on the quantities q 1 and q 2 supplied by each factory expressed by the function C ( q 1 , q 2 ) = 2 q 2 1 + q 1 q 2 + q 2 2 + 500 . The companies objective is to produce 200 units while minimizing pro- duction costs. How many units should be supplied by each factory? Mathematically, we are asking to minimize a function (namely the cost function C ( q 1 , q 2 )), given certain restrictions on the inputs - namely that we want precisely 200 units, or rather q 1 + q 2 = 200. Therefore, in mathematical terms, we interpret the problems as follows: Minimize the function C ( q 1 , q 2 ) = 2 q 2 1 + q 1 q 2 + q 2 2 + 500 subject to the constraint q 1 + q 2 = 200. This is not a standard minimization problem because we have very specific restrictions on the domain. This means we cannot just apply the derivative test and instead need to develop a new way to solve such...
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This note was uploaded on 10/28/2010 for the course MA 261 taught by Professor Stefanov during the Fall '08 term at Purdue University-West Lafayette.
- Fall '08