**Mathematical World**

Volume: 1;
1991;
187 pp;
Softcover

MSC: Primary 00; 01; 46; 49;

Print ISBN: 978-0-8218-0165-9

Product Code: MAWRLD/1

List Price: $28.00

Individual Member Price: $22.40

**Electronic ISBN: 978-1-4704-2469-5
Product Code: MAWRLD/1.E**

List Price: $28.00

Individual Member Price: $22.40

# Stories about Maxima and Minima

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*V. M. Tikhomirov*

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.

#### Table of Contents

# Table of Contents

## Stories about Maxima and Minima

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Table of Contents vii8 free
- Introduction ix10 free
- Part One. Ancient Maximum and Minimum Problems 114 free
- The first story Why Do We Solve Maximum and Minimum Problems? 316
- The second story The Oldest Problem—Dido's Problem 922
- The third story Maxima and Minima in Nature (Optics) 1932
- The fourth story Maxima and Minima in Geometry 2740
- The fifth story Maxima and Minima in Algebra and in Analysis 3750
- The sixth story Kepler's Problem 4760
- The seventh story The Brachistochrone 5568
- The eighth story Newton's Aerodynamical Problem 6578

- Part Two. Methods of Solution of Extremal Problems 7992
- The ninth story What is a Function? 8194
- The tenth story What is an Extremal Problem? 93106
- The eleventh story Extrema of Functions of One Variable 99112
- The twelfth story Extrema of Functions of Many Variables. The Lagrange Principle 109122
- The thirteenth story More Problem Solving 119132
- The fourteenth story What Happened Later in the Theory of Extremal Problems? 143156
- The last story More Accurately, a Discussion 179192

- Bibliography 187200
- Back Cover Back Cover1201