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inflexion point
In examining the graphs of differentiable real functions, it may be useful to state the intervals where the function is convex and the ones where it is concave.

A function $f$ is said to be convex on an interval if the restriction of $f$ to this interval is a (strictly) convex function; this may be characterized more illustratively by saying that the graph of $f$ is concave upwards or concave up. On such an interval, the tangent line of the graph is constantly turning counterclockwise, i.e., the derivative $f^{{\prime}}$ is increasing and thus the second derivative $f^{{\prime\prime}}$ is positive. In the picture below, the sine curve is concave up on the interval $(\pi,\,0)$.

The concavity of the function $f$ on an interval correspondingly: On such an interval, the graph of $f$ is concave downwards or concave down, the tangent line turns clockwise, $f^{{\prime}}$ decreases, and $f^{{\prime\prime}}$ is negative. In the picture below, the sine curve is concave down on the interval $(0,\,\pi)$.

The points in which a function changes from concave to convex or vice versa are the inflexion points (or inflection points) of the graph of the function. At an inflexion point, the tangent line crosses the curve, the second derivative vanishes and changes its sign when one passes through the point.

A graph may have points where the second derivative vanishes but does not change its sign when passed such a point; thus the first derivative is here changing “very slowly”. Such a point is sometimes called an undulation point. The graph of $x\mapsto x^{4}$ has the origin as an undulation point.
Since the sine function is $2\pi$periodic, the sinusoid possesses infinitely many inflexion points. Indeed, $f(x)=\sin x$; $f^{{\prime\prime}}(x)=\sin x=0$ for $x=0,\,\pm\pi,\,\pm 2\pi,\,\dots$; $f^{{\prime\prime\prime}}(x)=\cos x$, $f^{{\prime\prime\prime}}(n\pi)=\cos n\pi=(1)^{{n+1}}\neq 0$. Nonnullity of the third derivative at these critical points assures us the existence of those inflexion points.
Remarks
1. For finding the inflexion points of the graph of $f$ it does not suffice to find the roots of the equation $f^{{\prime\prime}}(x)=0$, since the sign of $f^{{\prime\prime}}$ does not necessarily change as one passes such a root. If the second derivative maintains its sign when one of its zeros is passed, we can speak of a plain point (?) of the graph. E.g. the origin is a plain point of the graph of $x\mapsto x^{4}$.
2. Recalling that the curvature $\kappa$ for a plane curve $y=f(x)$ is given by
$\kappa(x)\;=\;\frac{f^{{\prime\prime}}(x)}{[1+f^{{\prime}}(x)^{2}]^{{3/2}}},$ 
we can say that the inflexion points are the points of the curve where the curvature changes its sign and where the curvature equals zero.
3. If an inflexion point $x=\xi$ satisfies the additional condition $f^{{\prime}}(\xi)=0$, the point is said to be a stationary inflexion point or a saddlepoint, while in the case $f^{{\prime}}(\xi)\neq 0$ it is a nonstationary inflexion point.
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changes of an entry not visible
Hi unlord, I have today edited the entry ”inflexion point” but the changes have not been registered. Jussi
Re: changes not registered Hm...
That’s a new one, I’m not sure what to say. What happens if you make a trivial change like adding “test edit”? …
OK, I tried, and I got an error message that said the Parent article couldn’t be referenced. But I fixed that problem locally and I was able to edit the article. Can you delete the test edit I made to the last line? I think it should work now.
I’ll go into the database and fix the incorrect “parent” links globally.
changes of an entry not visible
Thanks unlord, Now the edit canges are normally registered. How about the diagrams made with the pstricks? For example in the entry ”inflexion point” the pstricks code does not work.
Re: pstricks
It certainly looks like it’s trying to render the article. I’ve asked here: https://trac.mathweb.org/LaTeXML/ticket/1708 so we should know soon.
pstrics images
Jac, it really seems that no pstricks images are now rendered. There are quite many such images here. Jussi
pstricks  at "proof of concept" stage
Jussi:
What I’ve least heard from the LaTeXML developers is as follows:
“Just to start getting you all excited for the various graphics goodies expected to land with the 0.8 release of LaTeXML, you can now enjoy a PSTricks proofofconcept demo in the LaTeXML showcase:
http://latexml.mathweb.org/editor (select ”PSTricks Graphics” from the dropdown menu at the bottomleft)
Thanks to Joe for providing the example from PlanetMath and to Bruce for getting the dust off the pstricks binding.”
I think the 0.8 release will be soon.
The example is:
So at least that one example works. I think we’ll get them all sorted out fairly soon! Joe