Talk:Critical point (mathematics)

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Mathematics rating:	Start Class	High Priority	Field: Analysis
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. 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)

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)

[edit] Critical vs. 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).

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)

[edit] Needs new image

Current image is: http://upload.wikimedia.org/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)

[edit] 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)