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Solving multivariate optimisation problems using inequalities

Published online by Cambridge University Press:  16 October 2017

Stephen Kaczkowski
Stephen Kaczkowski*
Affiliation:
South Carolina Governor's School, 401 Railroad Avenue, Hartsville, SC 29550 e-mail: kaczkowski@gssm.k12.sc.us
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Extract

Optimisation problems are among the most practical applications of calculus to everyday life, and a survey of exercises in various calculus textbooks will provide a teacher with many interesting scenarios for framing intriguing questions on this topic. Whether it is finding a container's dimensions that yield the least surface area for a given volume, or finding that ideal movie ticket price which will maximise a theatre's revenue, students can usually relate to these problems. Pólya in his book Plausible reasoning makes the following remarks about the attraction of extrema problems:

Problems concerned with greatest and least values, or maximum and minimum problems, are more attractive, perhaps, than other mathematical problems of comparable difficulty, and this may be due to a quite primitive reason. Every one of us has his personal problems. We may observe that these problems are very often maximum or minimum problems of a sort. We wish to obtain a certain object at the lowest possible price, or the greatest possible effect with a certain effort, or the maximum work done within a given time and, of course, we wish to run the minimum risk. Mathematical problems on maxima and minima appeal to us, I think, because they idealize our everyday problems. We are even inclined to imagine that Nature acts as we would like to act, obtaining the greatest effect with the least effort [1, p. 121].

Type
Articles
Information
The Mathematical Gazette , Volume 101 , Issue 552 , November 2017 , pp. 412 - 423
Copyright
Copyright © Mathematical Association 2017 

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References

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