Applications of inflection points

Question

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.

Answer 1

One option is to think about inflection points as places where growth begins to slow (going from concave up to concave down), or where growth begins to increase (going from concave down to concave up).

Answer 2

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.

Answer 3

I like using finance examples for these sorts of things. Everybody understands money. An inflection point could mean significant growth of a certain company, or it could mean that a market bubble is about to burst. For stock graphs, you can argue that its more important to know where the inflection points are than the minimum and maximum in order to maximize your returns. Such graphs are easy to find, and you can even tie in some engineering history with certain companies. Take a look at the stock value of a major engineering firm, and try to tie inflection points with real world events: natural disasters, major lawsuits, completion of landmark projects, etc...

Answer 4

In the context of maxima and minima, inflection points are need-to-know because they are false positives for the first derivative test. This motivates verifying that a critical point is actually an extremum.

Additionally, the Principle of Least Action is misnamed. It is actually the Principle of Stationary Action. Likewise, Fermat's Principle should be stated in terms of stationarity instead of minimum time. (Look at yourself in the concave surface of a spoon for an example. This example paraphrased from here.) It is not hard to construct an optical system whose solutions are variational inflections because the maxima or minima are forbidden by other features of the system.

See also the Feynman Lectures on Physics, Vol. 2, Chapter 19, especially p. 19-6, the paragraph starting "'One remark: I did not prove it was a minimum...'".

I'd make the (unjustified, sweeping) generalization that any physics using the calculus of variations cares about minima, maxima, and inflections.

Stepping up one dimension, saddle-points are one kind of false positive with utility. The method of steepest descent relies on finding saddle-points.

Answer 5

Two things spring to my mind: 1) inflection points occur where the first derivative is at a max or min, so the magnitude of the slope of the tangent is maximized. Therefore, any situation that is changing dynamically will benefit from a knowledge of it's inflection point which signifies the greatest rate of change. (ie on a plot of employment rates vs time, an inflection point would occur when the economy is growing most vigorously or collapsing most dramatically). 2) Since inflection points represent a change in concavity of a function, it represents a situation where the external forces are changing. (related to point 1 of course). At the traffic light example above, it's the moment where the foot comes off the gas pedal and moves to the brake.