Concept

Inflection point

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa. For the graph of a function of differentiability class C2 (f, its first derivative f, and its second derivative f'', exist and are continuous), the condition f'' = 0 can also be used to find an inflection point since a point of f'' = 0 must be passed to change f'' from a positive value (concave upward) to a negative value (concave downward) or vice versa as f'' is continuous; an inflection point of the curve is where f'' = 0 and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point. In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex''' is defined as a point where the tangent meets the curve to order at least 4. Inflection points in differential geometry are the points of the curve where the curvature changes its sign. For example, the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative f' has an isolated extremum at x. (this is not the same as saying that f has an extremum). That is, in some neighborhood, x is the one and only point at which f' has a (local) minimum or maximum. If all extrema of f' are isolated, then an inflection point is a point on the graph of f at which the tangent crosses the curve. A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing.

Official source

https://en.wikipedia.org/wiki/Inflection_point

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (32)

MATH-106(f): Analysis II

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables. ...

MATH-189: Mathematics

Ce cours a pour but de donner les fondements de mathématiques nécessaires à l'architecte contemporain évoluant dans une école polytechnique. ...

EE-556: Mathematics of data: from theory to computation

This course provides an overview of key advances in continuous optimization and statistical analysis for machine learning. We review recent learning formulations and models as well as their guarantees

Related lectures (32)

Extrema of Functions

Covers the discussion of local extrema, concavity, convexity, and inflection points in functions.

Applications of Differential Calculus

Explores applications of differential calculus, including theorems, convexity, extrema, and inflection points.

Local Extremum Conditions: n=2 and n=3

Explains local extremum conditions for n=2 and n=3, critical points, and stationary points.

Related publications (10)

Accelerated Stochastic Min-Max Optimization Based on Bias-Corrected Momentum

Lower-bound analyses for nonconvex strongly-concave minimax optimization problems have shown that stochastic first-order algorithms require a sample complexity of at least O(ε-4) to find an ε-stationary point. Some works indicate that this complexity can be improved to O(ε-3) when the stochastic loss gradient is Lipschitz continuous. The question of achieving enhanced convergence rates under distinct conditions remains open. In this work, we address this question for optimization problems that are nonconvex

2026

Decentralized Accelerated Nonconvex Minimax Optimization via Exact Diffusion

Exact Diffusion is a powerful algorithm for decentralized minimization problems over multi-agent networks, showing significant advantages over gradient-tracking methods, particularly in sparsely connected networks. Despite its success, this technique has remained largely unexplored in the context of decentralized minimax optimization, leaving it unclear whether its benefits extend to this setting. To address this gap, we propose a new algorithm that integrates Exact Diffusion with accelerated momentum for n

IEEE2026

Training Dynamics of Gradient Methods on Simple Neural Architectures

A central challenge in modern machine learning is to understand \emph{why} and \emph{how} gradient-based training of neural networks leads to models that generalize well and learn meaningful internal representations. This thesis approaches this challenge by studying the training dynamics of first-order methods on a sequence of models of increasing complexity, from linear predictors through linear networks to attention-only transformers. In each setting, the goal is to obtain a closed-form, end-to-end chara

EPFL2026

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Official source

https://en.wikipedia.org/wiki/Inflection_point

About this result

Ontological neighbourhood

Mathematics

Analysis: Calculus

Related courses (32)

MATH-106(f): Analysis II

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables. ...

MATH-189: Mathematics

Ce cours a pour but de donner les fondements de mathématiques nécessaires à l'architecte contemporain évoluant dans une école polytechnique. ...

EE-556: Mathematics of data: from theory to computation

Related lectures (32)

Extrema of Functions

Covers the discussion of local extrema, concavity, convexity, and inflection points in functions.

Applications of Differential Calculus

Explores applications of differential calculus, including theorems, convexity, extrema, and inflection points.

Local Extremum Conditions: n=2 and n=3

Explains local extremum conditions for n=2 and n=3, critical points, and stationary points.

Related publications (10)

Accelerated Stochastic Min-Max Optimization Based on Bias-Corrected Momentum

2026

Decentralized Accelerated Nonconvex Minimax Optimization via Exact Diffusion

IEEE2026

Training Dynamics of Gradient Methods on Simple Neural Architectures

EPFL2026

Related people (12)

Volkan Cevher

Volkan Cevher received the B.Sc. (valedictorian) in electrical engineering from Bilkent University in Ankara, Turkey, in 1999 and the Ph.D. in electrical and computer engineering from the Georgia Institute of Technology in Atlanta, GA in 2005. He was a Research Scientist with the University of Maryland, College Park from 2006-2007 and also with Rice University in Houston, TX, from 2008-2009. Currently, he is an Associate Professor at the Swiss Federal Institute of Technology Lausanne and a Faculty Fellow in the Electrical and Computer Engineering Department at Rice University. His research interests include machine learning, signal processing theory, optimization theory and methods, and information theory. Dr. Cevher is an ELLIS fellow and was the recipient of the Google Faculty Research award in 2018, the IEEE Signal Processing Society Best Paper Award in 2016, a Best Paper Award at CAMSAP in 2015, a Best Paper Award at SPARS in 2009, and an ERC CG in 2016 as well as an ERC StG in 2011.

Kimon Antonakopoulos

Riccardo Rattazzi

Riccardo Rattazzi was born in Novara (Italy) in 1964. He studied physics at the University of Pisa, where he received the Laurea cum laude in 1987, and at the Scuola Normale Superiore where he received the Diploma in Scienze and carried out graduate research in theoretical physics. After having been a post-doctoral research associate at the Lawrence Berkeley Laboratory, at Rutgers University and at CERN, in 1998 Riccardo obtained a permanent research position at the Istituto Nazionale di Fisica Nucleare in Pisa. From 2001 to 2006 he was a staff member at the Theory Division of CERN. In 2006 he was appointed professor of physics at EPFL.

Dragan Damjanovic

Dragan Damjanovic received BSc diploma in Physics from the Faculty of Natural Sciences and Mathematics, University of Sarajevo, in 1980, and PhD in Ceramics Science from the Department of Materials Science and Engineering, College of Earth and Mineral Sciences, the Pennsylvania State University (PSU) in 1987. From 1988 to 1991 he was a research associate in the Materials Research Laboratory at the PSU. He joined the Ceramics Laboratory, Department of Materials Science and Engineering, Ecole polytechnique fédérale de Lausanne in 1991. He is currently a "professeur titulaire", heads the Group for Ferroelectrics and Functional Oxides at the Institute of Materials and teaches undergraduate and graduate courses on structure and electrical properties of materials. The research activities include fundamental and applied investigations of piezoelectric, ferroelectric and dielectric properties of a broad class of materials.

Patrick Thiran

Patrick Thiran is a full professor in network and systems theory at the School of Computer and Communication Sciences at EPFL. He holds an electrical engineering degree from the Université Catholique de Louvain, Louvain-la-Neuve, Belgium, an M.Sc. degree in electrical engineering from the University of California at Berkeley, USA, and he received the PhD degree from EPFL, in 1996. He became an adjunct professor in 1998, an assistant professor in 2002, an associate professor in 2006 and a full professor in 2011. He was with Sprint Advanced Technology Labs in Burlingame, California, in 2000-01. His research interests are in communication and social networks, performance analysis and stochastic models. He is currently active in the analysis and design of wireless and PLC networks (scaling laws, medium access control), in network monitoring (network tomography, multi-layer networks), and data-driven network science. He also contributed to network calculus and to the theory of locally coupled neural networks and self-organizing maps. He served as an associate editor for the IEEE Transactions on Circuits and Systems in 1997-99 and for the IEEE/ACM Transactions on Networking in 2006-10. He is currently on the editorial board of the IEEE Journal on Selected Areas in Communication. He is/was on the program committee of different conferences in networking, including ACM Sigcomm, Sigmetrics, IMC, CoNext and IEEE Infocom. He was TPC chair of AMC IMC 2011 and CoNext 2012. He is a Fellow of the Belgian American Educational Foundation and of the IEEE. He received the 1996 EPFL Doctoral Prize and the 2008 Crédit Suisse Teaching Award.

Haoyuan Cai

Mikhail Chapochnikov

Mikhail Shaposhnikov was born in Sochi, Russia, in 1956. He got his degree from Moscow States University in 1979 and his PhD from the Institute for Nuclear Research of the Russian Academy of Sciences in 1982 for his work on the problem of baryon asymmetry of the universe in grand unified theories. He worked from 1982 till 1986 as a junior research scientist and from 1986 till 1991 as a senior research scientist at the Theory Division of the Institute for Nuclear Research of the Russian Academy of Sciences (Moscow). From 1991 till 1998 he worked as a staff member at the Theory Division of European Organisation for Nuclear Research (CERN, Geneva). In 1998 he was appointed as a Professor of Theoretical Physics at the University of Lausanne, in 1999 he became a Director of the Institute of Theoretical Physics of Lausanne University. Since October 2003 he is a Professor at EPFL, leading the Laboratory for Particle Physics and Cosmology.

Denis Gillet

Denis Gillet received the Diploma in Electrical Engineering from the Swiss Federal Institute of Technology in Lausanne (EPFL) in 1988, and the Ph.D. degree in Information Systems also from the EPFL in 1995. During 1992 he was appointed as Research Fellow at the Information Systems Laboratory of Stanford University in the United States. He is currently Maître d'enseignement et de recherche at the EPFL School of Engineering, where he leads the React research group. His current research interests include Technologies Enhanced Learning (TEL), Human Computer Interaction (HCI), Human Devices Interaction (HDI) and Optimal Coordination of Complex and Distributed Systems. Denis Gillet is affiliated at EPFL with the Center for Intelligent Systems and the Center for Digital Education.

Philippe Müllhaupt

Expertise Nonlinear Control (Geometry and Algebra)Classical DynamicsRoboticsStatistics Mission Provide rigorous technical and scientific training in nonlinear control, classical mechanics, and statistical theory Current Work Research and teaching in the fields of control theory, statistics, dynamical systems, and general mechanics. Philippe Mullhaupt received his Diplôme d Education Ingï¿½nieur | Electricitï¿½ 1993 – 1993 EPFL DEA | Automatique et Traitement du Signal 199

Giancarlo Ferrari Trecate

Expertise Automatic control, Dependable control, Neural-network Control, Machine learning, Microgrids, Hybrid systems Mission See the webpage of the DECODE (Dependable Control and Decision) group. Teaching & PhD PhD Students Laura Meroi, Julien Pallage, Nicolas Léo Pierre Kirsch, Simone Baratto, Daniele Martinelli, Riccardo Cescon, Jakob Nylöf, Sabri El Amrani Past EPFL PhD Students Pulkit Nahata, Mustafa Sahin Turan, Baiwei Guo, Jean-Sébastien Brouillon, Clara Galimberti, Andrea Martin, Muhammad Zakwan,

Related units (1)

Adaptive Systems Laboratory

Related concepts (22)

Curve sketching

In geometry, curve sketching (or curve tracing) are techniques for producing a rough idea of overall shape of a plane curve given its equation, without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features. The following are usually easy to carry out and give important clues as to the shape of a curve: Determine the x and y intercepts of the curve. The x intercepts are found by setting y equal to 0 in the equation of the curve and solving for x.

Stationary point

In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient is zero).

Hessian matrix

In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants".

Related exercises (2)

Spring Oscillator

A block of mass m is connected to the floor and the ceiling of a room of height h by two springs. The springs have elastic constants k₁ and k₂ and null rest lengths, as indicated on the diagram. Gravity acts on the block. We only consider the block’s vertical motion, along an x-axis pointing downwards with its origin on the ceiling. We neglect the block’s shape, as well as any friction.

Draw, on a diagram, all the forces acting on the block and give their expressions.
Find the differential equ

Asymmetrical pendulum

A pendulum is made up of a marble of mass m connected to a fixed point O with a massless string of constant length. A nail is placed at a distance d from point O, vertically below, such that the pendulum as lengthxa0L when it goes right and lengthxa0du2005+u2005L when it goes left (see diagram). The marble is placed such that the string makes an anglexa0α (α

Related notebooks (2)

Modeling Epidemics with Signal Processing Techniques

Epidemics and Signal Processing[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In [1]: import matplotlib.pyplot as plt import numpy as np import scipy.signal as sp import seaborn as sns sns.set() In [2]: plt.rcParams["figure.figsize"] = (14,4) The basic reproduction number[4]¶ The propagation mechanism of an epidemic such as the one caused by SARS-CoV-2 virus can be roughly modeled, at least in its inital phase, as a process in which each infected individ

Designing FIR Filters Using the Parks-McClellan Algorithm

The Parks-McClellan Filter Design Method[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

The Parks-McClellan (PM) algorithm is a clever application of advanced polynomial fitting techniques to the problem of FIR filter design. In this notebook we will explore the key ideas behind the method by considering the design of a simple Type-I lowpass filter. NOTE: This notebook uses interactive widgets. If the interactive controls do not work or show up, please fo