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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. Notorious examples include the functions $\cos(1/x)$ and $\sin(1/x)$ near $x=0$


and $\sin(e^{2x+9})$ near 0 and 1.

\begin{figure}\begin{center}\BoxedEPSF{ExpSin.epsf scaled 800}\end{center}\end{figure}

The latter has

\left\lfloor{{e^{11}\over\pi}-{1\over 2}}\right\rfloor -\lef...{{e^9\over\pi}-{1\over 2}}\right\rceil +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


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.

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.

© 1996-9 Eric W. Weisstein