Unimodal illustration

The figure has a clear maximum in . ${\displaystyle f(x)=1-x^{2}}$ ${\displaystyle x=0}$

A unimodal mapping or unimodal function in mathematics is a function with a unique (local and global) maximum such as, for example . ${\displaystyle f(x)=-x^{2}}$

definition

The precise definition is as follows:

A mapping of an interval in itself with is unimodal if there is such that for strictly monotonically increasing and for strictly monotonically decreasing . ${\displaystyle f\colon I\to I}$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle f(\partial I)\subset \partial I}$ ${\displaystyle m\in I}$ ${\displaystyle f(x)}$ ${\displaystyle x\leq m}$ ${\displaystyle x\geq m}$

From the definition it follows that the maximum function value is and that besides it has no further local maxima . ${\displaystyle f(m)}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle x}$

Examples

• A quadratic function with is unimodal. ${\displaystyle f(x)=ax^{2}+bx+c}$ ${\displaystyle a<0}$
• The entropy in information theory is unimodal.
• The negative of the absolute value function is unimodal.
• The tent image is unimodal.

literature

• Anušić: Dynamics of unimodal interval maps
• Bruin: Combinatorics of (Fibonacci-like) unimodal maps