# Maxima and Minima

## Applications

There are numerous practical applications in which it is desired to find the maximum or minimum value of a particular quantity. Such applications exist in economics, business, and **engineering**. Many can be solved using the methods of differential calculus described above. For example, in any manufacturing business it is usually possible to express profit as a function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. In other cases, the shape of a container may be determined by minimizing the amount of material required to manufacture it. The design of piping systems is often based on minimizing **pressure** drop which in turn minimizes required pump sizes and reduces cost. The shapes of **steel** beams are based on maximizing strength.

Finding maxima or minima also has important applications in **linear algebra** and **game theory**. For example, linear programming consists of maximizing (or minimizing) a particular quantity while requiring that certain constraints be imposed on other quantities. The quantity to be maximized (or minimized), as well as each of the constraints, is represented by an equation or **inequality**. The resulting system of equations or inequalities, usually linear, often contains hundreds or thousands of variables. The idea is to find the maximum value of a particular variable that represents a solution to the whole system. A practical example might be minimizing the cost of producing an **automobile** given certain known constraints on the cost of each part, and the **time** spent by each laborer, all of which may be interdependent. Regardless of the application, though, the key step in any maxima or minima problem is expressing the problem in mathematical terms.

## Resources

### Books

Abbot, P., and M. E. Wardle. *Teach Yourself Calculus.* Lincolnwood (Chicago), IL: NTC Publishing, 1992.

Allen, G.D., C. Chui, and B. Perry. *Elements of Calculus.* 2nd ed. Pacific Grove, CA: Brooks/Cole Publishing Co., 1989.

Larson, Ron. *Calculus With Analytic Geometry.* Boston: Houghton Mifflin College, 2002.

Lay, David C. *Linear Algebra and Its Applications.* 3rd ed. Redding, MA: Addison-Wesley Publishing, 2002.

Paulos, John Allen. *Beyond Numeracy, Ruminations of a Numbers Man.* New York: Alfred A Knopf, 1991.

Silverman, Richard A. *Essential Calculus With Applications.* New York: Dover, 1989.

J. R. Maddocks

## Additional topics

Science EncyclopediaScience & Philosophy: *Mathematics* to *Methanal trimer*Maxima and Minima - Applications