Unimodal function

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

Examples of unimodal function:


 * Quadratic polynomial
 * Logistic map
 * Tent map

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.