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Extremum

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An extremum is a maximum or minimum. An extremum may be local (a.k.a. a relative extremum; an extremum in a given region which is not the overall maximum or minimum) or global. Functions with many extrema can be very difficult to graph.

CosSin

Notorious examples include the functions cos(1/x) and sin(1/x) near x=0, plotted above.

ExpSin

Another pathological example is sin(e^(2x+9)) near 0 and 1, which has

|_(e^(11))/pi-1/2_|-[(e^9)/pi-1/2]+1=19058-2579+1=16480

extrema in the closed interval [0,1] (Mulcahy 1996).

See also

Global Extremum, Global Maximum, Global Minimum, Kuhn-Tucker Theorem, Lagrange Multiplier, Local Extremum, Local Maximum, Local Minimum, Maximum, Minimum

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972.Mulcahy, C. "Plotting and Scheming with Wavelets." Math. Mag. 69, 323-343, 1996.Tikhomirov, V. M. Stories About Maxima and Minima. Providence, RI: Amer. Math. Soc., 1991.

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Extremum

Cite this as:

Weisstein, Eric W. "Extremum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Extremum.html

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