Quod Erat Demonstrandum

2008/05/13

Definition of inflection point

Filed under: HKALE,Pure Mathematics — johnmayhk @ 1:22 下午

同工 William 君質疑:在 AL 1995 Pure Math (II) Q.9 中,為何 (0,0) 不是 inflection point(拐點,有譯:反曲點、迴折點)?

題目的方程是
f(x) = \frac{|x|}{(x + 1)^2} where x \ne -1.

y = f(x) 的圖像見下

單從圖像可見 “f^{''}(x) changes sign as x increases through x = 0“.
但 marking scheme 沒有包括 (0,0) 作為拐點。那麼,拐點的定義如何?

mathworld 的定義,inflection point 不是 relative maxima 或 relative minima。按此定義,上題的 (0,0) 就不是拐點,因 (0,0) 是 relative minimum point。據 math.com 的定義,拐點之處更要存在 tangent line。

但另一方面,據 wikipedia 的定義:"a point on a curve at which the second derivative changes sign",沒有提及它是否 relative extrema。但文中又有等價定義,包括:"a point on a curve at which the tangent crosses the curve itself",那麼是否暗示 tangent line 的存在?查 JSTOR 的論文,有一篇是 Darboux’s Theorem and Points of Inflection,因為不能 access,只能看到一小部分,其中在 google 的搜尋中顯示下面一句 “In fact, the definition allows extrema and points of inflection to coincide." 但這句究竟指什麼,不知道了。

我知,作為授課員,要找學術的書,不是在網上隨意亂 click;然而,我現在不能進大學的圖書館,也沒有網上買書的習慣(更大的理由是沒有時間也沒有足夠學力找正確的書),所以留待高人了。但相信學生認為最『真』的定義,應該是考評局定的。

拋一題:
Let f(x) = x^2\sin(\frac{1}{x}) for x \ne 0 and f(0) = 0.
Is (0,0) a point of inflection?

8 則迴響 »

  1. Differentiating x/(1+x)^2, we have

    1/(1+x)^2 – 2x/(1+x)^3, hence at x=0, the value of the derivative is 1.
    Thus for x/(1+x)^2, at x=0,the left-hand and right-hand derivative coincide, taking the value 1.

    f(x) = |x|/(1+x)^2 in the neighborhood of 0, so the left hand derivative becomes -1, while the right hand derivative remains as 1. Hence f(x) is not differentiable at x=0, so (0,0) can not be an inflection point.

    迴響 由 Ei — 2008/05/24 @ 7:52 下午 | 回覆

  2. Thank you for Ei’s reply..

    要問的是,(0,0) 不是 inflection point 的理由,究竟是

    1.(正如 Ei 所言)not differentiable at (0,0) 還是
    2. (0,0) is a minimum point?

    比方說,下圖顯示的點 P 明顯 not differentiable,它不是 extreme point,它的左右明顯有 concavity 的有變化,那麼它是否 inflection point?

    迴響 由 johnmayhk — 2008/05/25 @ 9:31 上午 | 回覆

  3. Wikipedia and math.com explicitly requires differentiability at the inflection point. This is commonly used by elementary calculus and engineering textbooks, so it is the most likely to be the one employed by HKEAA. Note that by requiring differentiability, we may also rule out extrema.

    The definition given by mathworld does not state conditions on differentiability at the inflection point, but superimposes another requirement that the point must not also be an extremum.
    That article from JSTOR explicitly denies “differentiability at the inflection point" a position among the list of necessary conditions. The consequences are that extrema may or may not also be inflection points. It is quite a plausible definition, to be honest.

    It is notable that in the latter two, the function in concern is not even explicitly required to be continuous at the inflection point. Messy conventions……….

    迴響 由 Ei — 2008/05/25 @ 7:01 下午 | 回覆

  4. Thank you Ei!

    For the definition mentioned in Wikipedia, actually I can’t figure out how the statement “… a point on a curve at which the curvature changes sign" implying “…f'(x) is at an extremum…" as an equivalence.

    I do want to read the article you found in JSTOR, would you mind sending me the pdf file through my E-mail: johnmayhk@yahoo.com.hk

    Million thanks!!! [Just take it easy if you are not available (or don’t want) to do so.]

    Also, there may be ‘something more’ in the definition of asymptote as mentioned in my previous post

    https://johnmayhk.wordpress.com/2007/11/02/alpm-asymptote/

    Ei, what’s your idea?

    Wish you’ll obtain excellent results in Exams!

    迴響 由 johnmayhk — 2008/05/26 @ 9:13 下午 | 回覆

  5. >johnmayhk@yahoo.com.hk

    Done.

    >For the definition mentioned in Wikipedia,…

    I agree with you it is very crudely addressed, as well as erroneous… but anyway, assuming that it tries to use just one definition, we’ll take the intersection of those alternative descriptions.

    迴響 由 Ei — 2008/05/27 @ 7:17 上午 | 回覆

  6. THANKS A LOT!!!!!!!! Ei.
    I’d received it.

    迴響 由 johnmayhk — 2008/05/27 @ 8:35 上午 | 回覆

  7. How about alpm 2003 Paper 2 Q.7?
    Why (- 1, 0) is both a relative maximum point and an inflection point and yet f is NOT differentiable at x = -1?

    迴響 由 mklaw — 2009/02/05 @ 5:05 下午 | 回覆

  8. Thank you mklaw. That’s an inconsistency in the definition from HKEAA?

    迴響 由 johnmayhk — 2009/02/05 @ 5:27 下午 | 回覆


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