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A necessary but not sufficient condition for a point of inflection is that $$f''(x)=0$$ If the second derivative is 0 and the point is not a point of inflection, Wikipedia tells me that is called an undulation point, which apparently means

a point on a curve where the curvature vanishes but does not change sign.

An example given is $f(x)=x^4$ at $(0,0)$.

What I do not understand is why $(0,0)$ is not classified as a local minimum? Surely for $f(x)=x^4$ the first derivative changes polarity either side of $(0,0)$.

Is there a better example of an 'undulation point' for a smooth function? Do 'undulation points' exist for more than one variable or is the condition that all partial derivatives are zero a necessary and sufficient one?

Gridley Quayle
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2 Answers2

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The point $(0,0)$ is a minimum point. It is also an undulation point. You are right that in some ways this is a poor example of an undulation point, since it also has other properties. On the other hand, this example does make the point easy to see, and it has an extremely easy formula.

A better example in some ways would be $(0,0)$ in the graph of $f(x)=x^4+x$. The point is still fairly noticeable but is not a minimum.

enter image description here

ADDED: I just took at look at the Wikipedia definition of undulation point, and it does give the example $f(x)=x^4$ in the text. However, it also has the example $y=x^4-x$ in a graph later in the article. This is the same as my example but reflected in the $y$ axis. It seems Wikipedia wanted to have it both ways.

Rory Daulton
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  • That is a better example thanks. I'm still confused as to why (0,0) is an undulation point for $f(x)=x^4$ though. Doesn't the sign of the derivative change either side of (0,0)? – Gridley Quayle Jan 06 '15 at 18:41
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    @GridleyQuayle: Yes, the derivative changes sign, but that is irrelevant to the definition of undulation point. The second derivative (and therefore the curvature) do not change sign, and that is what is relevant. – Rory Daulton Jan 06 '15 at 19:03
  • If you go to the Wikipedia page for 'Inflection Point' and go to the section 'a necessary but not sufficient condition' the example is at the end of the first paragraph. – Gridley Quayle Jan 06 '15 at 21:24
  • @GridleyQuayle: Ah, yes, I see, you are right. I shall edit my answer accordingly. I referred to the graph later in the article. – Rory Daulton Jan 06 '15 at 22:21
  • @GridleyQuayle [OFF-TOPIC PING] Please do not delete well-formed questions if they have been answered. They may still be useful for a future visitor. – AlexR Jan 11 '15 at 23:43
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    As a future visitor I thank You! – uhu23 Aug 17 '16 at 21:01
  • I too, as a future visitor, want to thank you. – ankit Aug 17 '17 at 13:27
  • @Gridley Quayle, I just want to add a minor thing here, and try to answer your question as to exactly why the example of $x^4$ is included in the definition of an undulation point. When you see the graph of $x^2$, it just touches the origin and you can say that the tangent is horizontal at just one point. Now, minutely observe the graph of $x^4$ or any even powered $x$ say, $x^{2n}$ (n=2,3,4,..). You will see that the curve seems to touch the axis at more than one point, or grazes through the axis. The tangent does not seem to be horizontal at just one point. continue – Stranger Forever Jun 13 '20 at 02:24
  • @Gridley Quayle continued from above Therefore, by the definition of undulation point, the curve loses it's concavity around the point $0$. That's it, and you are done !! Mathematics always talks to you, it's just that sometimes we aren't ready to listen :) – Stranger Forever Jun 13 '20 at 02:25
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Who said it's not a local minimum? It certainly is.

I don't know how you would want to define "undulation point" in several variables. One problem is that there isn't just one "curvature".

Robert Israel
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  • this got me thinking too, .. Since there are multiple ways to look at the curvature with respect to the variables present in multivariable calculus, can we not define undulation points separately for each variable? (ofcourse, if it has one). And if all the variables agree at some particular point that becomes the universal undulation point.. note that the word universal is just made up by me – Stranger Forever Jun 13 '20 at 02:32