Talk:Inflection point

Old isolated post
I have expanded this, and corrected a few errors including that the second derivative at a point of inflection can be non-zero. Also added the term 'saddle-point' for 'non-stationary point of inflection'. And made clear that in principle there is no limit to how high an order of derivative you might need to go to if you are using that method. 158-152-12-77 01:38, 24 August 2005 (BST)

Inconsistent definition of Saddle Point
The definition in this article differs substantially from the saddle point article. One or both need to be fixed to show the correct definition (or both definitions, if both are correct). I have never heard of the def in this article, but am familiar with the surface def in the separate saddle point article. However, that doesn't mean this def isn't also correct, as many terms have multiple meanings. StuRat 19:16, 24 August 2005 (UTC)


 * Your graphs are excellent - they add a lot to the article. The definition of 'saddle point' in the saddle point article is incomplete, because it does not cover the case of a function of a single variable. This is covered properly in the Mathworld definition, namely "A point of a function or surface which is a stationary point but not a extremum" (i.e. not a local extremum).
 * 158-152-12-77 21:41, 25 August 2005 (BST)


 * I have amended saddle point article.
 * 158-152-12-77 00:11, 26 August 2005 (BST)


 * Thanks. StuRat 23:18, 25 August 2005 (UTC)


 * The allusions in the present article are still not clear. The article currently has "An example of a saddle point is the point (0,0) on the graph y = x³. The tangent is the x-axis, which cuts the graph at this point."  But there is no way for the reader to understand how this relates to inflexion points.
 * For example, why don't we also write "An example of a local minimum is the point (0,0) on the graph y = x²." It's also true, and also lacks an explanation of how it relates (if at all) to the present article....
 * —DIV (120.17.88.168 (talk) 00:51, 28 September 2017 (UTC))
 * The use of the term "saddle point" for a univariate function is uncommon, although correct, and defined in Saddle point. Nevertheless, generally, this term is used for univariate functions only when talking of functions of any number of variables, including one variable. Thus there is no reason, and it is confusing, to use it here. I have fixed this. D.Lazard (talk) 07:37, 28 September 2017 (UTC)

Font problem
In four places towards then end of the article, when the first derivative is mentioned, it's impossible for me to see the apostrophe after the "f", so I thought the function itself was being referred to, and I got really confused. I really wish I could fix this myself, but I have no idea how.


 * The only way I can see to make it more visible is to add spaces:

HOW IT NOW APPEARS: f'(x)

WITH SPACES ADDED : f ' (x)


 * Should we do that ? StuRat 21:34, 2 December 2006 (UTC)

- No, probably not. One option (besides leaving it as it is) would be to force/trigger the TEX-style mathematical rendering. As I recall, this was typically done by including a character that cannot be rendered in pure HTML (e.g. a negative space) — but I cannot find a link to that advice at the moment, so maybe things have changed in the meanwhile. Here are some links: m:Help:Displaying_a_formula, Help:Displaying_a_formula, Manual_of_Style/Mathematics, Rendering_math. Anyway, other than that you might consider that mathematically we should be using a prime (symbol), and not a typewriter apostrophe, and that is a change that could be implemented just with HTML. —DIV (120.17.88.168 (talk) 01:17, 28 September 2017 (UTC))

Definition vs. Property
From the definitions: a point on a curve at which the tangent crosses the curve itself. This is a property of inflection points, but in no way defines them: tangents can cross the curve in many points, most of which, in some cases, are not inflection points. See Tangent.

Diego.pereira 20:49, 10 March 2007 (UTC)


 * I think you're misreading it: if it's an inflection point x=a, then we want the tangent line at the point a to cross the curve at the point a. --Cheeser1 (talk) 05:00, 19 November 2007 (UTC)


 * A tangent line by definition never crosses a curve. According to the Webster definition, a tangent line meets a curve or surface in a single point if a sufficiently small interval is considered. First Known Use in 1594. A tangent line cannot be constructed at any point of inflection or saddle point. 166.249.129.10 (talk) 01:09, 9 June 2012 (UTC)


 * I have moved the previous posy because it was placed in the middle of an older post. This is not allowed by the policies.
 * The first assertion of the post is wrong. It is worth to mention that the author of the post has vandalized tangent by introducing there his wrong wp:original research (now reverted).
 * D.Lazard (talk) 09:50, 9 June 2012 (UTC)


 * All the assertions are correct. Is it called vandalizing when others edit articles and editing when you edit articles? Lazard, you are a Dimwit! There is reference that proves the first assertion is true. See Webster definition of tangent point. Also, you keep reverting the tangent article after I edit it. You need to STOP doing this. It is wrong as it currently stands. Now if you don't like the definition, that is your problem. But mathematics is not about what you like or dislike. 166.249.132.163 (talk) 12:00, 9 June 2012 (UTC)
 * I recall you that personal attack are forbidden by Wikipedia policy and may cause to block you to edit. Moreover your assertion that all your assertions are correct is not a valid argument to convince anybody. D.Lazard (talk) 12:59, 9 June 2012 (UTC)

Graph of y=x^3 is wrong
The graph here resembles y=x^3 near the origin, but it doesn't pass the vertical line test like the x^3 function does. This doesn't affect what is being demonstrated, however, but nonetheless for the purposes of accuracy I believe the either figure should be replaced with a true x^3 graph, or that its caption contain the correct equation (unknown to me) for the graph shown. --unsigned


 * The first graph in the article is y=x^3 and is correct, it should be tangent to the x axis at 0. You may be talking about the cubic root. Oleg Alexandrov (talk) 03:42, 17 August 2007 (UTC)


 * I believe he's talking about how the second graph is not a function, Oleg. At x=4, notice how there are two y-values. I was about to say the same myself but I saw it was already here. --unsigned


 * Just plot x^3 + x instead of the rotated x^3 graph, which is not a function. --unsigned —Preceding unsigned comment added by 198.240.130.75 (talk) 14:39, 6 March 2008 (UTC)

Okay, first of all, a graph's not being a function has nothing to do with it's validity in demonstrating something mathematically. Circles aren't functions, so do you want them removed from wikipedia? Secondly, the caption doesn't claim that the graph is that of y = x^3, it claims that it is a graph of y = x^3 ROTATED. Nicklink483 (talk) 19:59, 28 September 2008 (UTC)

Important Change Needed
A location where the "tangent line at the point a crosses the curve at point a" is not necessarily an inflection point.

The definition of inflection point is that the curvature changes from positive to negative or negative to positive at the point. That is, there exist $$c, d$$ such that $$f''(x)$$ has opposite signs on the intervals $$(c,a)$$ and $$(a,d)$$, Consider the function $$f(x) = \left\{ \begin{array}{c l} x^{5}\sin(\frac{1}{x}) & x\ne0\\ 0 & x=0 \end{array} \right. $$

This function is twice differentiable on $$(-\infty, \infty)$$. The tangent line at $$x=0$$ is $$y=0$$ and crosses the curve there. However, the sign of $$f''(x)$$ changes infinitely often in all intervals of the form $$(c,0)$$ or $$(0,d)$$.

To correct the page, the statement should be taken out of the definition section and restated later as "If $$f$$ has an inflection point at $$a$$, then the tangent line at $$a$$ crosses the graph at $$a$$." —Preceding unsigned comment added by 64.59.248.162 (talk) 20:31, 10 March 2008 (UTC)


 * How do you come up with f' (0) = 0? Oli Filth(talk) 20:40, 10 March 2008 (UTC)
 * $$f'(0)=\lim\limits_{h\to 0} \frac{f(h)-f(0)}{h}=\lim\limits_{h\to 0}\frac{h^5\sin(1/h)}{h}=\lim\limits_{h\to 0}h^4\sin(1/h)=0$$ since $$|\sin(1/h)|\le1$$.
 * Ah yes.
 * However, presumably your example should be $$f(x) = x^4\sin(1/x)$$, as in your current example, we have $$f(x) = f(-x)$$, hence the tangent can only touch the curve at $$x=0$$, not cross it.
 * However, with $$f(x) = x^4\sin(1/x)$$, we have that $$f^{\prime\prime}(x) = -f^{\prime\prime}(-x)$$. Therefore, this example also satisfies a further test for an inflection point (namely that $$f^{\prime\prime}(x + \epsilon)$$ and $$f^{\prime\prime}(x-\epsilon)$$ should have opposite signs in the region of an inflection point at x).
 * Oli Filth(talk) 21:50, 10 March 2008 (UTC)


 * In any neighborhood of zero there are points a<0<b such that f(a)f(b)<0 . That's good enough for me. --Cheeser1 (talk) 21:36, 10 March 2008 (UTC)


 * Actually, I used the fifth power so that the function is twice differentiable. The comment that in any neighborhood of zero, there are points $$a<0<b$$ such that f(x)f(b)<0 is very interesting,  but does not make an inflection point according to the accepted definition.  —Preceding unsigned comment added by 64.59.248.162 (talk) 23:08, 10 March 2008 (UTC)
 * Which is...? --Cheeser1 (talk) 20:28, 11 March 2008 (UTC)


 * If we accept the definition that an inflection point of a graph is a point where concavity changes and we take the function:
 * $$f(x) = \left\{

\begin{array}{c l} x^{3} & x<0\\ x^{3}+1 & x\ge0 \end{array} \right. $$


 * at $$x=0$$ the concavity changes but the statement is not true: "If $$f$$ has an inflection point at $$a$$, then the tangent line at $$a$$ crosses the graph at $$a$$.", because there is no tangent line at $$x=0$$. —Preceding unsigned comment added by 194.63.137.68 (talk) 09:57, 11 October 2009 (UTC)

Suggestion for a change to definition
Not all points of inflection involve a change from concave upwards to concave downwards. Also in a closed curve or looping curve (a circle, for example), the curve changes from being concave upwards to concave downwards without there being any point of inflexion.

The second sentence of the definition requires changing. It says:

"The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa."

I think you should replaced this with the following:

"Typically this is when an open curve changes from being concave upwards to concave downwards or vice versa."

Stated as such it doesn't sound like the curvature (whether upwards or downwards) should be part of the definition at all. —Preceding unsigned comment added by 79.97.237.43 (talk) 11:06, 11 July 2008 (UTC)


 * Your statement "the curve changes from being concave upwards to concave downwards without there being any point of inflexion" is sheer nonsense. The portion of the circle above the diameter is concave downwards - there is no change. The portion of the circle below the diameter is concave upwards. This means there must be an inflection point at x=r and x=-r. In fact there is because a tangent with defined gradient cannot be constructed at either of these points. 166.249.135.45 (talk) 22:14, 9 June 2012 (UTC)

Thanks for the analogy!
The analogy with the steering wheel in this article is really good -- I use that concept all the time now when doing calculus homework. I suggest that for the future it's left in there and not removed, as it makes a complex concept far easier to understand. —Preceding unsigned comment added by 71.210.95.34 (talk) 01:46, 25 October 2008 (UTC)

Analyticity and the equivalent formulations of inflection point
The article states three formulations of the definition of an inflection point and claims they are equivalent. In general, they are not equivalent as previous comments indicate. A function f for which f' has an extremum at which f' does not change from increasing to decreasing or vice versa is the function f:R→R defined by f(0)=0 and f(x) = exp(-1/x^2)(sin(1/x))^2 for x≠0. A function g for which there is a point where the tangent line crosses the graph but at which g' does not have an extremum is g:R→R defined by g(0) = 0 and g(x) = sgn(x)f(x) for x≠0 where f is the function from the previous example. Both of these examples are infinitely many times continuously differentiable. Interestingly, for real analytic functions the three formulations are equivalent because the zeros of a power series are isolated.Jalongi (talk) 23:17, 6 April 2009 (UTC)

a related stuff
fxx(x0,y)=0 implies f achieves local extreme at x0 along x. fyy(x,y0)=0 implies f achieves local extreme at y0 along y. then what does fxy(x0,y0)=0 mean? cross inflection point? Jackzhp (talk) 21:57, 20 April 2010 (UTC)

Tangent crosses the curve?
The article states: ...a point on a curve at which the tangent crosses the curve at that point.

The above line should be removed because it is incorrect. If the "tangent" crosses the curve at a point, it is no longer a tangent.

Webster defines tangent as: meeting a curve or surface in a single point if a sufficiently small interval is considered 

Although the derivative is defined for the cubic at x=0, it does not represent the slope of a tangent. There is no tangent at x=0. Consider the case of the circle derivative f'(x)= -x/sqrt(r^2-x^2). Although the derivative is defined, if x=-r or x=r, the derivative is undefined whereas a vertical tangent exists at both points: (-r,0) and (r,0). These are exceptions to the rule and the reason we call such points inflection points. 114.225.82.109 (talk) 10:09, 8 June 2010 (UTC)

In the context given, it is perfectly reasonable to describe an inflection point as a point at a curve at which the tangent crosses the curve at that point. The context is a differentiable function, which should be carefully explained as a real-valued function of a single real variable whose derivative exists in an interval. In that context, the tangent should be defined, not using Webster, as the line with slope equal to the derivative at a point which intersects the curve at that point. This line "crosses" the curve if and only if in every neighborhood of that point, there are points of the curve above the line and points of the curve below the line. Above and below are well-defined concepts since any non-vertical line in the plane divides the plane into three subsets: the line itself, the half-plane above the line, and the half-plane below the line. Points whose y-coordinate is greater than the y-coordinate of the point's projection onto the line are above; points whose y-coordinate is lesser than the y-coordinate of the point's projection onto the line are below. If the tangent exists at a point, and the curve is concave downward in the left half of a neighborhood of that point, and the curve is concave upward in the right half of the same neighborhood, points to the left of the point in that neighborhood will be below the line, and points to the right of the point in that neighborhood will be above the line.

Curves which do not happen to be functions may also be considered to have curvature and possibly inflection points, but the definitions require a bit more topology than is usually needed when restricting attention to curves which are differentiable functions. TheDean (talk) 15:36, 17 April 2021 (UTC)

possible error
in the section of equivalent forms it says: "a point (x, y) on a function, f(x), at which the first derivative, f′(x), is at an extremum, i.e. a minimum or maximum. (This is not the same as saying that y is at an extremum)."

If you take f(x)=x, then f'(x)=1 is always at a maximum. but f has no inflection points. I think it should be changed from "minimum or maximum" to "local minimum or local maximum" - which is the usual meaning of extremum point anyway... —Preceding unsigned comment added by 79.177.105.99 (talk) 13:53, 14 May 2011 (UTC)

Use in higher dimensions
For higher dimensional functions one can still define the boundary between concave and convex regions, though it's no longer a point. In a 3-dimensional graph, I've heard people use "inflection line" before. Is there standard terminology for this? Do people use "inflection curve" or "inflection region" (when the graph is in >3 dimensions)? --Bequw (talk) 21:32, 1 August 2012 (UTC)


 * I've noticed that in the car body design industry, they call them "inflection lines", despite them clearly being a curve, not a line. StuRat (talk) 21:34, 1 August 2012 (UTC)


 * Apparently, you are talking of the points where one of the two principal curvatures is null. These points are called parabolic points, and the standard terminology in geometry is thus "curve of parabolic points". There are other remarkable curves and points on smooth surfaces, see Umbilic point and Ridge (differential geometry). D.Lazard (talk) 09:28, 2 August 2012 (UTC)
 * I have forgot another important curve on smooth surfaces: the line of curvature. About the use of "line" to name some curves, one must be aware that the modern meaning of "line" is an abbreviation of "straight line", like "curve" is an abbreviation of "curved line". Moreover, in other languages, like French, "straight line" has been abbreviated by the equivalent of "straight" ("droite" in French), and the equivalent of "line" (ligne, in French) denotes (straight) lines as well as curves. This explains why "line" is yet used, in some cases, to denote curves.D.Lazard (talk) 11:18, 2 August 2012 (UTC)


 * D.Lazard mentions "one of the two principal curvatures".


 * Why only two? 2601:200:C000:1A0:31AA:B51:BC4C:5F5B (talk) 01:22, 9 September 2022 (UTC)

2nd Derivative unneeded in plotting the cubic - read on!
I discovered this in my college Calculus class. It was party due to this discovery, that I scored somewhat well in that class, as mathematics are not my strong point. I do however, have a natural talent in seeing patterns that warrant further exploration. My calculus professor was blown away by this discovery, and strongly urged me to try publishing it. I would have, except that I'm a very honest sort, and so researched it further. The ONLY other reference to this, was by a Professor of Louisiana State University - in 1929! Below, I have quoted just the preface to Irby C. Nichols' paper on the subject to show that I am not crazy, this is real, and light should be further shed on this in the mathematical community.

MY version is this: To find an inflection point, you only need the nearest local minimum and maximum points of any curve. If you plot a new line between the two, it will ALWAYS intersect the curve at the inflection point. Go ahead, try it before you read any further; I'll wait. I'm not fluent enough in calculus to provide the mathematical proof of this, but Professor Nichols WAS.

"The present paper offers two suggestions which should be- time-saving both to high school and college teachers of mathematics in their plotting of cubic equations. The author has used them with profit a good many years and, although he does not recall ever having seen them treated in any text on algebra or calculus, he feels that they possess merit sufficient to warrant special attention.

Familiarity with the popular definitions of maximum and minimum points and a point of inflection is assumed. It is also assumed that we can obtain the first derivative of a cubic and can apply it in obtaining its maximum and minimum points. No knowledge of the second derivative is required: the point of the present brief treatise is to show that the second derivative is not needed in plotting a cubic."

"Plotting the Cubic Irby C. Nichols Mathematics News Letter, Vol. 3, No. 5. (Jan., 1929), pp. 18-21. Mathematics News Letter is currently published by Mathematical Association of America."

(I discovered this through JSTOR, which hosts a copy.) — Preceding unsigned comment added by Karmana (talk • contribs) 03:44, 8 October 2012 (UTC)


 * I agree that this is an interesting and curious property, and that it is true. I congratulate you to have discovered it. But I disagree with Nichols that it is useful: If you want use it to compute (numerically) the coordinates of the inflection point, this is much more complicated than with the second derivative. I doubt that you will be able to do such a computation. If you want to use it to locate graphically the inflection point, you will see that the angle between this line and the tangent at the inflexion point is small. Thus, for a reasonably accurate localization of the inflection point, you will need a very high accuracy of the plotting. As the computation of the inflection point is only useful for having a correct plotting by computing only a small number of points on the cubic, you have some kind of vicious circle: you need an accurate plotting to get basic information on the plotting. :Nevertheless, as you have a reference, it could be discussed if this result is worth to be included in WP. If it is, cubic function would be a better place, because this result does not generalize to higher degree functions. --D.Lazard (talk) 10:20, 8 October 2012 (UTC)

undulation
can someone add an actual example of undulation point? JMP EAX (talk) 14:15, 15 August 2014 (UTC)

And how would one define undulation for a C0 function, e.g. for a polyline? The concept is obviously used for the latter (without giving a def) e.g. in. JMP EAX (talk) 14:19, 15 August 2014 (UTC)
 * The graph of y=xn has an undulation point at the origin, when n is an even integer greater than 2. As far as I know, undulation points are defined only for C3 curves (this is needed for having a curvature, which is defined and has a constant sign, in a small neighborhood, on both sides of the point). It is possible that generalizations of the definition exists for less regular curves, but they are not standard. D.Lazard (talk) 15:44, 15 August 2014 (UTC)
 * Why do you believe that C2 curves cannot have curvature defined (and continuous).
 * Fact: They do. 2601:200:C000:1A0:31AA:B51:BC4C:5F5B (talk) 01:29, 9 September 2022 (UTC)

Labeling of Graph of Cubic upwards downwards
An earlier version of the graphic has purple labels 'f(x) curve concave downwards' and 'f(x) curve concave upwards.' The article mentions this nomenclature, as well as 'concave' and 'convex' more frequently seen beyond school mathematics. As the purple labels were adjusted to reflect this latter usage, and the former usage parenthesized, it appears that the distinction between upwards and downwards was lost. Both occurrences in parentheses now read 'downwards.'

Perhaps I have misunderstood the intent, but shouldn't the label on the LHS read: concave (downwards); and the label on the RHS read: convex (concave upwards). Or if spacing is an issue, 'concave (down)' and 'convex (concave up)' respectively? Zimri2 (talk) 04:05, 11 December 2019 (UTC)

=Serious error=

f(x)=x^4 has f(x) continuous and f(0)=0, but that doesn't make x=0 an inflection point, it is a local extrema. If f'(a)=f''(a)=...f^{(m-1)}(a)=0 and f^(m)(a)!=0, then a local extrema if m is even, and it is an inflection point if m is odd.

Thomaso (talk) 07:03, 18 June 2021 (UTC)
 * Please, read the article before saying that it is erroneous: this is shortly discussed in the lead, and section discusses this with details, with a title that emphasizes what you say. The case of $$f(x)=x^4$$ is explicitly considered in this section. D.Lazard (talk) 09:02, 18 June 2021 (UTC)

It may seem obvious, but it is not
The section Definition begins as follows:

"Inflection points in differential geometry are the points of the curve where the curvature changes its sign."

Unfortunately, it is not at all clear what this means if a curve is not necessarily of class C2 at the point in question.

I hope someone knowledgable on this subject can fix this omission. 2601:200:C000:1A0:31AA:B51:BC4C:5F5B (talk) 01:19, 9 September 2022 (UTC)