Recently, I was teaching maxima, minima and inflection points to first year engineering students. I motived extrema by giving practical examples of optimization problems, but when a colleague asked me later about applications of inflection points, I didn't have a ready answer beyond sketching graphs.
An initial search online didn't seem to turn up many appropriate* applications, although this MSE question: What is the purpose of defining the notion of inflection point? provided some assistance. However, its focus is on inflection points in pure mathematics.
Does anyone here have any ideas on motivating inflection points,
especially to practically-minded engineering students? When is knowing the location of inflection points important in applications?
$*$ By appropriate, I mean suitable for an introductory calculus class. I see that inflection points show up, for instance, when discussing Bézier curves. However, this topic might take me too far afield.
Best Answer
For simple calculus of single variable, any example with variable whose second variable changes sign (or, equivalently, whose first derivative changes behavior between growing and falling) will do.
Non-physics example: it can be said that in some parts of the world (perhaps also in the world as whole) population has reached an inflection point. This means that, while it still may be growing, its growth is slowing down, while before the inflection point it was speeding up (in some stages it might have even been an exponential growth).
A more advanced (and physics) example would be following:
When doing the calculus of variations, one seeks to make the action $S$ stationary, which is in classical mechanics: \begin{equation} S=\int L dt \end{equation} or, in a field theory: \begin{equation} S=\int \mathcal L d^4x \end{equation} where $L$ and $\mathcal L$ are lagrangian and lagrangian density respectively.
This is often called the principle of least action. However, the action does not have to be minimal, it can also be also be maximal (for example when finding the timelike geodesics), so this principle is also (a bit more correctly) called the principle of extemal action. However, it suffices that variation of the action $\delta S$ vanishes, so the action only needs to be stationary (principle of stationary action).
Therefore, the action can also be in an inflection point, although I can't think of any specific examples at the moment.