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Throughout the history of mathematics, maximum and minimum problems have played an important role in the evolution of the field. Many beautiful and important problems have appeared in a variety of branches of mathematics and physics, as well as in other fields of sciences. The greatest scientists of the past---Euclid, Archimedes, Heron, the Bernoullis, Newton, and many others---took part in seeking solutions to these concrete problems. The solutions stimulated the development of the theory, and, as a result, techniques were elaborated that made possible the solution of a tremendous variety of problems by a single method.
This book presents fifteen "stories" designed to acquaint readers with the central concepts of the theory of maxima and minima, as well as with its illustrious history. This book is accessible to high school students and would likely be of interest to a wide variety of readers.
In Part One, the author familiarizes readers with many concrete problems that lead to discussion of the work of some of the greatest mathematicians of all time. Part Two introduces a method for solving maximum and minimum problems that originated with Lagrange. While the content of this method has varied constantly, its basic conception has endured for over two centuries. The final story is addressed primarily to those who teach mathematics, for it impinges on the question of how and why to teach. Throughout the book, the author strives to show how the analysis of diverse facts gives rise to a general idea, how this idea is transformed, how it is enriched by new content, and how it remains the same in spite of these changes.
Ancient maximum and minimum problems:
1. Why do we solve maximum and minimum problems?
2. The oldest problem - Dido's problem.
3. Maxima and minima in nature (optics).
4. Maxima and minima in geometry.
5. Maxima and minima in algebra and in analysis.
6. Kepler's problem.
7. The brachistochrone.
8. Newton's aerodynamical problem. Methods of solution of external problems.
9. What is a function?
10. What is an external problem?
11. Extrema of functions of one variable.
12. Extrema of functions of many variables. Lagrange's principle.
13. More problem solving.
14. What happened later in the theory of extremal problems.
15. The fifteenth story, or rather, a discussion.
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