# 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.

Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group.

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.

## KEY TERMS

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Derivative

—The rate at which a function changes with respect to its independent variable. Geometrically, this is equivalent to the slope of the tangent to the graph of the function.

Domain

—The set, or collection, of all the first elements of the ordered pairs of a function is called the domain of the function.

Function

—A set of ordered pairs. It results from pairing the elements of one set with those of another, based on a specific relationship. The statement of the relationship is often expressed in the form of an equation.

Range

—The set containing all the values of the function.