Talk:Inflection point

message: 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)
thread_title:  Font problem 

message: 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)
replies: text: 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)
thread_title:  Suggestion for a change to definition 

message: 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)
thread_title:  Thanks for the analogy! 

message: 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)
thread_title:  Analyticity and the equivalent formulations of inflection point 

message: 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)
thread_title: a related stuff

message: 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 <straight line tangent to a curve>
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)
thread_title:  Tangent crosses the curve? 

message: 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)
thread_title:  possible error 

message: 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)
replies: text: 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)
thread_title:  2nd Derivative unneeded in plotting the cubic - read on! 

message: 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)
thread_title:  Labeling of Graph of Cubic upwards downwards 

message: 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)
replies: text: 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)
thread_title: Serious error

message: 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)
thread_title:  It may seem obvious, but it is not