Talk:Critical point (mathematics)

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critical points of F(x,f(x))=1/2(f(x))^2+1/2x^2-1/6x^3 —The preceding unsigned comment was added by 69.155.40.117 (talk • contribs) .

When dF/dx = 0 i.e. f(x) df/dx + x - 1/2 x^2 =0 --Salix alba (talk) 16:19, 27 January 2006 (UTC)

Are singular points critical?

In Adams Calculus (ed 7, but also the current ed 10) a critical point is only a point with gradient = 0, and singular points (function is not differentiable/gradient does not exist) are excluded. As Adams is cited, currently at the very least the citation is incorrect. Stewart Calculus however defines a critical point to include those where the gradient does not exist. As there is no clear definition in the standard textbooks this leaves the question which definition wikipedia should use. But regardless, citing Adams and using another definition does not seem to be the way to go. — Preceding unsigned comment added by Tasar (talk • contribs) 08:46, 30 September 2024 (UTC)

Critical v. Stationary

"It is also called a stationary point."

Isn't that only valid when the derivative is zero (i.e. and not when it is undefined, like the article suggests)? —The preceding unsigned comment was added by 87.194.47.191 (talk) 09:10, 24 February 2007 (UTC).

Quite right. Cerberus (talk) 15:01, 28 September 2016 (UTC)

This is to add to the example of x + 1/x. This article says that '0' is a critical point. A crtical point, however MUST be in the domain of the function and '0' is NOT in the domain of

x + 1/x

I think a better example would be to use a piecewise defined function that has a hole in the graph but is still defined there maybe slightly above or below the hole. Consider:

f(x) = { x + 1 when x != 3;

      { 2     when x  = 3;

3 is IN the domain and is also a critical number. Any thoughts?

To answer the above question the article is correct in stating that a critical number is any number 'c' in the domain of a function f(x) such that a function f'(c)=0 of f'(c) DNE —Preceding unsigned comment added by 209.148.171.223 (talk) 05:35, 11 November 2007 (UTC)

Needs new image

Current image is: http://up.wiki.x.io/wikipedia/commons/6/60/Stationary_vs_inflection_pts.gif

The current image showing critical points is inadequate, this page needs a new image.

The image shows points where the tangent line is zero, however it neglects points where the tangent line is not differentiable. It also shows inflection points, which are not critical points.

So I request someone create a new image, which shows both points where the tangent line is zero, and not differentiable (preferably an example of various types of these, such as when there is a break, when there is a vertical asymptote, when the tangent line is vertical) And I request the image not show points which are irrelevant to the article, such as inflection points, as they add nothing, and only cause confusion for someone who does not understand what a critical point is.

MiseryEverAfter 10:10, 14 November 2007 (UTC)

First Example is Wrong

For a number to be a critical point, either the derivative must be zero, or the derivative must be undefined/not exist AND it must be in the domain of the function. The point x=0 for the function f(x) = x - 1/x is NOT a critical point because x=0 is not in the domain of f.

A function such as g(x) = x^(2/3) is better -- the derivative does not exist at zero but zero is still in the domain of the function. —Preceding unsigned comment added by 130.126.108.104 (talk) 19:45, 13 February 2008 (UTC)

I decided to change it. -- Marc 130.126.108.211 (talk) 22:04, 13 February 2008 (UTC)

Need new definition

The term "critical point" is used in the definition of a Misiurewicz point, but there is no definition for that term in complex dynamics. Can you define it, please? --  Denelson83  14:43, 27 July 2010 (UTC)

Picture

The main thing that's wrong with the picture is that it calls points on the graph and not in the domain critical, contradictory to the definition. Nijdam (talk) 08:29, 21 September 2011 (UTC)

Nice work the new picture, but it is not appropriate, as critical points are points in the domain. Nijdam (talk) 21:16, 19 October 2011 (UTC)

Critical vs. stationary

First of all, critical point is defined only for differentiable functions and this is not said presently in this article. Secondly, the notion of critical point is fundamental notion is differential geometry, and the definition that is given here is not the specialization of the general notion to the elementary case. It is possible that some elementary textbooks give the definition of this article, but this only means that their authors have not understood differential geometry.

The notion of critical point is basically related to differentiable maps between manifolds. These are the points where decreases the rank of the Jacobian matrix of the map. I acknowledge that this definition is not convenient for this article. But its adaptation to plane curves is: given a plane curve, the critical points of the projection on the x-axis are the points with vertical tangent; the points that are critical for the projection on the y-axis, are those with an horizontal tangent, which are the stationary points, when the curve is the graph of a function. In other words, the critical points are exactly those where the implicit function theorem does not applies. Therefore, the notion of "critical point of a function" does not exist except for implicit functions; at this level, it exists only for curves.

IMO, the article needs to be completely rewritten along these lines, but keeping (at least for the first sections) its elementary level.

D.Lazard (talk) 16:41, 2 October 2013 (UTC)

Contradictory introduction

Against the above background: we might be trying to achieve too many things in the opening sentence, which doesn't make sense currently. It says (and I have italicized for emphasis), "In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0 or undefined." The problem is that, if the function is differentiable, there is no such thing as points at which the derivative is undefined.

Mebden (talk) 13:04, 31 October 2016 (UTC)

The problem comes from the fact that there are two different definitions (IMO, essentially one in mathematics, the other in educational mathematics). I have edited the lead and one example for removing the contradiction and clarifying the point. D.Lazard (talk) 14:12, 31 October 2016 (UTC)

Bad grammar

The phrase "does not applies" seems to be bad grammar.

 Fixed D.Lazard (talk) 14:21, 10 October 2013 (UTC)

Parsing errors

There were parsing errors that seem to have disappeared with the commas placed outside of the LaTeX math environment. However, this formatting contradicts MOS:MATH#PUNC. I don't know what's the best thing to do here. Isheden (talk) 09:54, 21 October 2013 (UTC)

Alternative definition of critical point of a differentiable map

There is an alternative definition for a critical point in the case of a differentiable map between two manifolds: a critical point of f is a point of R^m where the rank of the Jacobian matrix of f is less than n. It is found, for example, in Lafontaine, "An Introduction to Differential Manifolds".

A feature of this convention is that all points are considered critical when m < n (which I explicitly mentionned in my edit). This is not a mistake: this is intended to be so, and useful for some applications. (For example, Sard's theorem still holds with this definition, but becomes stronger: it gives as a corollary that every submanifold of codimension at least 1 has measure 0.)

Lafontaine is far from being alone in using this convention. I currently have no example of other *books* using this convention, but Googling "critical point" turns up several examples of this definition in lecture notes, e.g. http://www-personal.umich.edu/~wangzuoq/437W13/Notes/Lec%2009.pdf

On these grounds, I suggest re-reverting my latest edit. Ilia Smilga (talk) 03:27, 11 February 2016 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Critical point (mathematics)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This is a fundamental topic, but the current brief treatment is confused, and distracted by side issues. A critical point is a point where the derivative fails to be surjective. This is a subtle concept which needs to be introduced through examples. Geometry guy 00:40, 21 May 2007 (UTC) I don't agree with this. First, the definition should not use words like "surjective" as they are not needed. Second, the definition is that a critical value of a real-valued function of a real variable, f, is a number c in the domain of f at which the derivative, f'(x), is zero or undefined. Third, the value is the domain element, and the critical point is the associated (x,y). The existing article has this wrong. Dr. Mike DrMWEcker (talk) 04:13, 9 April 2008 (UTC)

Last edited at 04:13, 9 April 2008 (UTC). Substituted at 01:56, 5 May 2016 (UTC)