“Maximum”的概念、定义、习题解题方法及翻译-数学辞典

单词

Maximum

释义

Maximum

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 simplestOrder Statistics.

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

For a function which is Continuous at a point , a Necessary but notSufficient 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 maximafrom 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 , theExtremum Test succeeds under more general conditions than the Second Derivative Test.

See also Critical Point, Extremum, Extremum Test, First Derivative Test, Global Maximum,Inflection Point, Local Maximum, Midrange, Minimum, Order Statistic,Saddle Point (Function), Second Derivative Test,Stationary Point

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.

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