Is there significance to the maximum curvature of a Cartesian function?

Question

The curvature $\kappa$ has been derived for functions in Cartesian coordinates as $$\kappa=\frac{y''}{(1+y'^2)^{ \frac{3}{2}}}.$$ I noticed that the maximum value of the curvature typically corresponds a sort of$\dots$"vertex" looking point of a function where the tightness peaks as with $x=\dfrac{1}{2}\ln\big(\dfrac{1}{2}\big)$ of $y=e^x,$ and I am wondering if there is information on that concerning a formal name for this point of maximum curvature or if there are any special properties of this point.

Answer 1

It's sometimes called a "vertex", and with that naming, probably the most famous use is in the four vertex theorem. See wikipedia's article on the subject.

Of possible related interest: A generalization of the "Four vertex theorem"