Critical points of second derivative

[songoku]

TL;DR

By finding the critical points of f' (x) (point where f'(x) = 0 or f'(x) is undefined) and constructing the sign diagram for f', we can find point of relative maxima, relative minima and horizontal inflection of f

Using the same method for f", we can also find point where the concavity of f will change

If the sign on the sign diagram of f" changes from positive to negative or from negative to positive, this means the critical points of f" is non-horizontal inflection of f

But what about if the sign does not change? Let say f"(x) = 0 when $x = a$ and from sign diagram of f", the sign on the left and right of $a$ is both positive, what information can we get regarding point $x=a$ ? Is there a certain term to name that point?

Thanks

 

[Office_Shredder]

If the second derivative hits zero but doesn't change sign, that means the third derivative changed sign.

https://en.m.wikipedia.org/wiki/Third_derivative

Vs

https://en.m.wikipedia.org/wiki/Second_derivativ

As you can probably tell from the lack of content, people don't care that much about the third derivative.

 

[FactChecker]

The curvature is in the same direction on both sides of a point that has a zero curvature only at that point. I have never heard a mathematical name for that. I would say that the function is concave but not strictly concave (or convex but not strictly convex) around that point. Of course, if the sign of the second derivative changes, it is an inflection point.

 

[FactChecker]

As you can probably tell from the lack of content, people don't care that much about the third derivative.

The third derivative of position is "jerk", which can be significant sometimes.

 

[Office_Shredder]

The curvature is in the same direction on both sides of a point that has a zero curvature only at that point. I have never heard a mathematical name for that. I would say that the function is concave but not strictly concave (or convex but not strictly convex) around that point. Of course, if the sign of the second derivative changes, it is an inflection point.

If f'' is strictly positive except at a single point, I think f is strictly convex in an interval around that point. Convexity isn't defined by the second derivative being positive, it's just a useful test.

Equivalently and easier to think about, the function $x^3$ is strictly increasing even though the derivative is zero at one point.

 

[FactChecker]

If f'' is strictly positive except at a single point, I think f is strictly convex in an interval around that point. Convexity isn't defined by the second derivative being positive, it's just a useful test.

I stand corrected. Thanks.

 

[songoku]

Thank you very much Office_Shredder and FactChecker