We discuss the real world and visual interpretation of a point of inflection.

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Some astute students may have noticed that the algebraic rules for concavity seemed fairly familiar, and indeed, were eerily similar to the rules for increasing and decreasing. This makes sense, after all we did deduce that concave up is akin to an “increasing rate of change” and concave down was akin to the “decreasing rate of change”. But this begs the followup question: what about where the second derivative is zero? For the first derivative we saw that these points, called critical points, had the potential to be maximums or minimums. What about where the second derivative is zero?

These are the kinds of questions mathematicians would ask, to see how deep the rabbit hole goes. It turns out, that we have a similar outcome here as well! These values, where the second derivative is zero, have the potential to be points of inflection. We will cover the algebraic definition and how to find them in another segment, but for now let’s see what it means to be a point of inflection.

Remember that concavity can be thought of as bending the graph either upward (concave up) or downward (concave down). But obviously functions can have segments that are concave up, and other segments that are concave down, so what does the transition look like between the two? Let’s consider a graph as an example:

We can see from point A to B is concave down, and from B to C it is concave up. But at point B, the function is momentarily between concavity types - transitioning from concave down to concave up. This is exactly what a point of inflection is: a transition point between concavity - either from concave up to concave down, or vice versa.

One way to think about points of inflection is...
A point that transitions between a maximum and a minimum. A point that exists between two zeros. A point that transitions between two different types of concavity. Your arch nemisis, awaiting a moment of weakness to strike you down in the prime of your mathematical education!

I can hear you saying “Ok, big deal math nerd... But, do these points mean anything in a real world setting?” The answer, (surprisingly?) is yes!

Inflection points typically represent a sudden and paradigm-shifting event, something that significantly alters the trajectory of whatever they are modeling. For example, when COVID-19 hit, many countries shut down significant portions of their international interactions, including trade. This resulted in depressed economic activity world-wide. But prior to this, most economies had been steadily improving after the 2008 economic collapse. The result was a sudden shift from a recovering and improving world economy starting around 2011, to depressed and even negative growth when COVID-19 hit at the start of 2020. But what does this transition look like if we modeled it with a(n overly simplistic) graph? It would look something like this:

Now we can see that the graph is concave up as the recovery initially takes over, but then it shifts to concave down as the lockdown depressed the economy. That transition point between the recovery and the recession (the point that represents when COVID hit) is an example of an inflection point. It represents a severe paradigm shift that changed the economy from improving and trending toward a progressively better economy, to one that was hamstrung by a global pandemic.

This is just one example, and on a large scale, but individual companies can have similar events. For example: the invention of the rewritable CD crushed the floppy disk industry, or the invention of the smartphone preceded the takeoff of the app-writing market. Indeed, computers themselves have been responsible for many of these inflection points in individual businesses, markets, and even some national economies. In fact, these kinds of sudden market shifts due to new technology, unforeseen global phenomena (like pandemics or natural disasters) or sudden appearance or collapse of major competitors is so recognizable in this way that they are often referred to as inflection points or “inflection point events” in business.

Inflection points in businesses can be recognized as...
Sudden significant change that fundamentally changes the growth or viability of a company or product, A minor change that can help or harm a company’s growth, to a lesser but noticable degree. Nothing, there is no business or market interpretation of inflection points. Pandamonium. The end of days. The appocalypse is upon us!

We have introduced the idea of inflection points and their geometric interpretation - as transition points that represent major shifts in a business or economic model that fundamental change how that business or economy is doing: either from doing well to doing poorly or the other way around. Importantly, transition points don’t tend to represent simple shifts in the business landscape that a business may or may not need to adapt. Rather inflection points tend to be major moments that almost certainly make or break a business.