# Unimodal function: Wikis

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# Encyclopedia

In mathematics, a function f(x) between two ordered sets is unimodal if for some value m (the mode), it is monotonically increasing for xm and monotonically decreasing for xm. In that case, the maximum value of f(x) is f(m) and there are no other local maxima (i.e. there is one mode as the name indicates).

Examples of unimodal functions:

A function f(x) is "S-unimodal" if its Schwartzian derivative is negative for all $\ x \ne 0$.

In probability and statistics, a "unimodal probability distribution" is a probability distribution whose probability density function is a unimodal function, or more generally, whose cumulative distribution function is convex up to m and concave thereafter (this allows for the possibility of a non-zero probability for x=m). For a unimodal probability distribution of a continuous random variable, the Vysochanskii-Petunin inequality provides a refinement of the Chebyshev inequality. Compare multimodal distribution.

In computational geometry if a function is unimodal it permits the design of efficient algorithms for finding the extrema of the function.[1]

## References

1. ^ Godfried T. Toussaint, "Complexity, convexity, and unimodality," International Journal of Computer and Information Sciences, Vol. 13, No. 3, June 1984, pp. 197-217.