Finding Transition Points Zero-Crossings, Maxima, Minima and Inflections

• Published in: Journal of the Washington Academy of Sciences Vol. 108; no. 4; pp. 45 - 58
• Main Author: Baclawski, Kenneth
• Format: Journal Article
• Language: English
• Published: Washington Academy of Sciences 01.12.2022
• ISSN: 0043-0439; 2573-2110

A function has a transition point when some property changes at the transition point. Such points represent important features of a function. We examine three kinds of transition points: sign, monotonicity and convexity. A sign transition point, usually called a zero-crossing, is a place where the function changes from negative to positive or vice versa. At a monotonicity transition point the function changes from increasing to decreasing or vice versa. At a convexity transition point the function changes from being convex to concave or vice versa. When the function is continuous these three kinds of transition points are roots, maxima/minima, and inflection points, respectively. In this note we specify algorithms for finding the three kinds of transition points. We formally prove that the algorithms always converge, and we compare these algorithms with other algorithms. In the proof of the algorithm for inflections, we give a geometric interpretation of the mediant and its connection to convexity.