Quod Erat Demonstrandum

2008/05/13

Definition of inflection point

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

$f(x) = \frac{|x|}{(x + 1)^2}$ where $x \ne -1$.

$y = f(x)$ 的圖像見下

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。

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 | 回應

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 | 回應

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 | 回應

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/

Wish you’ll obtain excellent results in Exams!

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

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 | 回應

6. THANKS A LOT!!!!!!!! Ei.

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

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 | 回應

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

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