Minimum

The smallest value of a set, function, etc. The minimum value of a set of elements is denoted or, and is equal to the first element of a sorted (i.e., ordered) version of . For example, given the set 3,5, 4, 1, the sorted version is 1, 3, 4, 5, so the minimum is 1. The Maximum and minimum are the simplestOrder Statistics.

A continuous Function may assume a minimum at a single point or may have minima at a number of points. A GlobalMinimum of a Function is the smallest value in the entire Range of the Function, whilea Local Minimum is the smallest value in some local neighborhood.

For a function which is Continuous at a point , a Necessary but notSufficient condition for to have a Relative Minimum at is that be a CriticalPoint (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 minimafrom Maxima. 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 , theExtremum Test succeeds under more general conditions than the Second Derivative Test.

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.

Brent, R. P. Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ: Prentice-Hall, 1973.

Nash, J. C. ``Descent to a Minimum I-II: Variable Metric Algorithms.'' Chs. 15-16 in Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 186-206, 1990.

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.

• Elliptic Helicoid
• Elliptic Hyperboloid
• Elliptic Integral
• Elliptic Integral of the First Kind
• Elliptic Integral of the Second Kind
• Elliptic Integral of the Third Kind
• Elliptic Integral Singular Value
• Elliptic Integral Singular Value k1
• Elliptic Integral Singular Value k2
• Elliptic Integral Singular Value k3
• Ellipticity
• Elliptic Lambda Function
• Elliptic Logarithm
• Elliptic Modular Function
• Elliptic Paraboloid
• Elliptic Partial Differential Equation
• Elliptic Plane
• Elliptic Point
• Elliptic Pseudoprime
• Elliptic Rotation
• Elliptic Theta Function
• Elliptic Torus
• Elliptic Umbilic Catastrophe
• Ellison-Mendès-France Constant
• Elongated Cupola