Solving constrained optimization problems with Lagrange multipliers:
Part 1: Preparation
Introduction:
This activity will guide you through a graphical exploration of the
method of Lagrange multipliers for solving constrained optimization problems.
The central ideas will be illustrated with the following example.
The preparatory questions below are to be completed prior to the in-class
portion of the activity.
Preparatory questions:
(1) Describe the family of level surfaces of the objective function .
(2) Does the objective function obtain a maximum value in -space? Give an explanation which relates your answer to the family of level surfaces of .
(3) Describe some subsets of -space on which does attain a maximum value. (Try to produce a variety of examples with qualitative differences.)
(4) Describe some subsets of -space on which does not attain a maximum value. (Again try to produce a variety of examples with qualitative differences.)
(5) Suppose is a regular parameterization of a space curve. Use the chain rule to determine the rate of change of (with respect to ) as is evaluated along the curve. (In other words, compute .) Express your answer explicitly in terms of the gradient of and the velocity of .
(6) From the origin, in which direction should you go to increase the value of as rapidly as possible? Explain.
(7) Describe the set of all directions in which one can move from the origin so that will increase. Explain. (Feel free to discuss the subtleties of slightly different interpretations of this exercise with your instructor.)
(8) In questions (6) and (7), if the origin is replaced by an arbitrary space point do the answers change?
(9) Describe the solution set of the first constraint equation from problem (*) above.
(10) Describe the solution set of the second constraint equation .
(11) In general what types of sets can occur as the intersection of ? Comment on the problem of optimizing a continuous function in each case.
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