Talk:Stationary point

A picture would be much wanted to show the points of minimum, the points of maximum, and the inflection points. Oleg Alexandrov 13:05, 23 December 2004 (UTC)

Agreed. I added said pics, what do you think ? StuRat 22:56, 27 August 2005 (UTC)

Could we please explain the difference between a stationary point and a Critical point, the latter of which this article does not seem to link to? 137.205.139.228 19:00, 29 April 2007 (UTC)

First paragraph of articles states "An equivalent definition is where the derivative of the function equals zero (known as a critical number)."

This suggests: f'(c) = 0 <==> c is a critical number, which is not true (reverse is not true)

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What about linked extremums?

--User:Vanished user 8ij3r8jwefi]] 13:42, 13 March 2006 (UTC)

Astronomical use

There really should be a comment about the use of the term in Astronomy. CFLeon 23:31, 17 May 2006 (UTC)

Such as ? StuRat 05:06, 18 May 2006 (UTC)

When the Earth passes an outer planet, it appears to stop and then reverse for awhile, then stop and go the first way. Those are called the stationary points of an orbit. CFLeon 22:37, 18 May 2006 (UTC)

Ok, this is a completely unrelated usage of the term, so I suggest you create a new article named stationary point (astronomy). Just click on the link to create the article. StuRat 02:59, 19 May 2006 (UTC)

Section 3.1 "Example" begins by discussing f' and points x1, x2, etc. with no mention of f or a picture or definition of the points. Is a picture missing? Thanks. EJR 19:37, 17 June 2007 (UTC)

I just noticed this too. I replaced them with examples of functions. HexTree (talk) 13:52, 11 November 2014 (UTC)

Anyone else agree this is confusing?

The first sentence in the current version is:

"a stationary point is an input to a function where the derivative is zero (equivalently, the gradient is zero): where the function "stops" increasing or decreasing (hence the name)."

But the function doesn't stop increasing or decreasing if the stationary point is an inflection point. Perhaps the sentence should be amended to be "...the DERIVATIVE of the function stops increasing or decreasing (hence the name)."

Please sign your posts with 4 tildes ~ ~ ~ ~ (no spaces). I have no idea whether you posted this 10 years ago or in October 2013 (latest modification). First, you are wrong in asserting that the function "doesn't stop increasing or decreasing" at a stationary point which is ALSO an inflection point. It is perfectly possible for both the first derivative and the second to be zero at a point on the function. At the point the slope is zero by definition, and hence by definition it DOES "stop" changing. OTOH, I agree that not only is the article confusing, but it surely must be wrong. It is wrong either by stating that the stationary point is a point on a curve (meaning that in n-dimensional space, it is a point in that space (n coordinates) OR by stating it is the "input" (whatever that is!) to a function. Is that supposed to mean "an element of the domain of the function"? The same definition can't cover both. This article needs to be cleaned up, it seems to have been patched together who don't understand what a function is (a map between sets). A point (element) of input is not a point on the map. Indeed, a point on the map or on the function is shorthand for the point representing the input elements and the output elements, which may be n- and m- dimensional respectively hence the point is m+n dimensional (or a subset thereof).Abitslow (talk) 01:00, 20 December 2013 (UTC)

Although badly written, the article is correct. "Badly written" because of the use of "input" instead of "point or value in the domain of the function (or map)". This trend of using uncommon terminology instead of the mathematically correct one is very common in articles on elementary mathematics. It is apparently based on the strange idea that avoiding correct technical terminology makes thing easier to understand. "Badly written" also because it could be made clearer that a stationary point is not a point of the graph of the function but that stationary points may easily be recognized by looking on the graph. D.Lazard (talk) 10:23, 20 December 2013 (UTC)

Sources

especially the difference between "critical point" and "stationary point" needs a source --92.203.47.202 (talk) 16:33, 4 August 2011 (UTC)

Critical vs. stationary

I have modified the lead to clarify the difference between stationary point and critical point, in order to follow the terminology of differential geometry, which is the standard for this question. Nevertheless, the remainder of the article needs to be rewritten and sourced in accordance with this edit.

By the way, above post on stationary points in astronomy is a good illustration of the subject: these stationary points are exactly the critical points of the projection of the orbit of a planet on the celestial sphere.

D.Lazard (talk) 16:06, 2 October 2013 (UTC)

The two body paragraphs attempting to distinguish between stationary and critical points made absolutely no sense to me, so I removed them. Anyone is welcome to put back in a clearer distinction. Ted.tem.parker (talk) 17:26, 4 July 2018 (UTC)

What is the actual definition?

I removed this from the lede, since it doesn't have a reference, and furthermore the grammar is unclear.

"it is also defined as the point that is not differentiable (the derivative does not exist) or the points is undefined"

Is this article about critical points or stationary points? Are they synonyms? It seems to me that "critical point" is a less specific term, e.g., it would describe a phase transition.

Arided (talk) 13:39, 14 September 2018 (UTC)

In fact, in advanced mathematics these terms are formally equivalent, but correspond to different point ov view. (In elementary textbook, the terminology varies as "critical point" is often considered as a synonymous of "singular point"). A point is stationary if the value of the function is almost constant near the point, that is if all partial derivatives are zero. In other words, it is stationary when one moves in the domain of the function. On the other hand, a point is critical, if, when moving in the codomain, the nature of the fiber (the inverse image of a point by the function) changes of nature. In the standard case of differentiable functions on a manifold, this occurs when the partial derivatives are all zero, that is when the point is stationary. D.Lazard (talk) 14:25, 14 September 2018 (UTC)

Definition

The lead sentence says a stationary point is a point in the domain, but the graphs show it as a point on the surface. If only one of these is correct, the other should be clarified (in the text or in the caption). If both are valid uses, the lead sentence should be modified to reflect this. Loraof (talk) 14:31, 11 April 2017 (UTC)

I've fixed the discrepancy, with a source. Loraof (talk) 20:02, 12 April 2017 (UTC)

Local maxima and minima

The article incorrectly conflates the definitions of local maxima and maximal turning point. These are distinct. f has a local maximum at a if f(x) ≤ f(a) for all x near a. (See, for example, M. Spivak, Calculus, Chapter 11, or more or less any standard calculus text.) Even for C¹ functions this can occur without the derivative changing sign. For example f(x) = 0. But more generally the terms local maximum and local minimum do not depend on the differentiability of the function and the article would be better if this were made clearer. I.Doust 27 February 2024. — Preceding unsigned comment added by 129.94.176.103 (talk) 23:15, 26 February 2024 (UTC)

The word "isolated" was omitted in section § Turning points. This is not a reason for removing a whole paragraph, when it suffices to add this word. D.Lazard (talk) 11:42, 27 February 2024 (UTC)