Talk:Concave function
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This page states that a twice-differentiable function is concave if and only if (iff) the second derivative is negative. I don't think this is correct.
A twice-differentiable function is STRICT concave if and only if the second derivative is negative.
Use a constant function as an example. By the simpler definition, a constant function (or a linear function) is always concave (but not strictly concave). However, the second derivative is ZERO, not negative.
- You are right; I corrected the text. --NeoUrfahraner 14:17, 1 Mar 2005 (UTC)
I think the "if and only if" for strict concavity needs to be relaxed though. Or the "negative" should be changed to "negative definite" or something similar.
In other words, take -x^4. This function is clearly STRICT concave. However, its second derivative AT THE ORIGIN is 0. Thus, a function is strict concave if its second derivative is less than or equal to zero with EQUALITY ONLY at the origin. That sounds like negative definite. <?>
If you use the "definiteness" terms, I think this extends well into multiple variables as well.
Just a thought.
- OK, corrected again --NeoUrfahraner 11:14, 2 Mar 2005 (UTC)
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[edit] Merging with convex function?
Merging looks like a good idea to me, but one should merge this article in convex function and not viceversa I think. Also, convex function looks like a rather well-written article, so hopefully the merged version will not be worse than what it is now. In short, if anybody is willing to merge, that person should be willing to take the necessary time to do a good job. Otherwise I would oppose a merger. Oleg Alexandrov (talk) 12:12, 24 October 2005 (UTC)
- See also Talk:Convex function. Oleg Alexandrov (talk) 00:29, 30 December 2005 (UTC)
[edit] Incorrect definition?
"The definitions of convex and concave functions given here appear to be incorrect. What is said to be a concave function is a convex function and what is said to be a convex function is a concave function.
I have always found those terms confusing. The Chinese characters for the two terms have the shapes that they describe, so it is easy to recognise. The charactor for convex (or concave downward) is a protrusion on top of a square, and the one for concave (or concave upward) is a depression on the top side of a square."
Sorry it seems that I posted the above by mistake. I pressed "Save" and did not realise that my comment is already posted. I did not mean to edit the page but only wanted to ask the author the page to check carefully. I have not learned how to delete the paragraphs I added at the end of the item, which look ugly. Could you delete them? Please check your definitions.
Ziheng Yang
- Would you mind making an account if you would like to contribute more? It is better to keep in touch that way.
- As far as I can tell, everything is correct. Concave up is the same as convex, meanining a function of this form" \_/
- Concave down is the same as concave, meaning a function of this form:
---- / \ / \
- Other comments? Oleg Alexandrov (talk) 18:27, 30 December 2005 (UTC)
[edit] Ok, so what about linear functions?
What happens when you apply this definition to linear functions? This article does not make mention of this case. --Stux 17:34, 16 February 2006 (UTC)
- What is the problem? Linear functions are concave, but not strictly concave. --NeoUrfahraner 18:45, 16 February 2006 (UTC)
- Ok I sort of see it being sense mentioned here, at least in terms of linear transformation functions, but nowhere is it explicitly mentioned as such and briefly explained. I, personally am still unclear as to why it's either (I can see that if it's convex it'll be concave too), since this article stipulates:
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If f(x) is twice-differentiable, then f(x) is concave iff f ′′(x) is non-positive.
- Linear functions are clearly twice-differentiable and f''(x) = 0, of course, and this is ... Oh I just answered my own question! Nevermind, Thanks! But I still think it needs to be more explicitly mentioned in these articles. --Stux 22:36, 16 February 2006 (UTC)
[edit] Convex on every subinterval
"Equivalently, f(x) is concave on [a, b] if and only if the function −f(x) is convex on every subinterval of [a, b]."
I feel like this line might be glomming together two separate points. Are both of the following correct?:
1) A function f is concave an [a,b] iff -f is convex on [a,b].
2) A function f is concave (convex) on [a,b] iff it is concave (convex) on every /proper/ (was this word left out of the original?--it makes the statement less trivial) subinterval of [a,b].
If these are both true, maybe they should replace the current language. Also, the placement of this part suggests that the statement is restricted to continuous functions. Is that desired? --Dchudz 15:48, 25 July 2006 (UTC)
