The largest value of a set, function, etc. The maximum value of a set of elements is denoted or , and is equal to the last element of a sorted (i.e., ordered) version of . For example, given the set , the sorted version is , so the maximum is 5. The maximum and Minimum are the simplest Order Statistics.

A continuous Function may assume a maximum at a single point or may have maxima at a number of points. A Global Maximum of a Function is the largest value in the entire Range of the Function, and a Local Maximum is the largest value in some local neighborhood.

For a function which is Continuous at a point , a Necessary but not Sufficient condition for to have a Relative Maximum at is that be a Critical Point (i.e., is either not Differentiable at or is a Stationary Point, in which case ).

The First Derivative Test can be applied to Continuous Functions to distinguish maxima from Minima. For twice differentiable functions of one variable, , or of two variables, , the Second Derivative Test can sometimes also identify the nature of an Extremum. For a function , the Extremum Test succeeds under more general conditions than the Second Derivative Test.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 14, 1972.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Minimization or Maximization of Functions.''
Ch. 10 in *Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England:
Cambridge University Press, pp. 387-448, 1992.

Tikhomirov, V. M. *Stories About Maxima and Minima.* Providence, RI: Amer. Math. Soc., 1991.

© 1996-9

1999-05-26