What is a point of inflection?

All of us have been through moments when we feel something has to change. A moment when we feel we need to change our behaviour or an attitude at risk of failure. From a mathematical point of view, things are not that so different. In order to simpler explain what a point of inflection is, I will use a graph which shows us the profits of an enterprise. In the horizontal axis (called x-axis) we have the passage of time and in the vertical axis (called y-axis) we will find the profits.

After analysing the graph, we can notice this enterprise goes through a good moment since as time goes by its profits are always increasing. However, if you have a look at point a, it seems something has happened. Actually, at that point something has changed concerning the behaviour of the enterprise profits. Although there was a constant increase of profits, the graph does not always behave the same way. On the left of the point a profits were less and less increasing. If that kept on going like that, the profits would decrease. Fortunately, something changed within the enterprise regarding that same point. So, on the right of the point a we can notice that the profits began to increase more and more. That point is called the point of inflection.

How do you recognise a point of inflection?

If we attentively look at the graph, we will notice that on the left of the point `a` the concavity turns down while on the right of the point `a` the concavity turns up. Therefore, the point of inflection is exactly placed where both concavities of the graph change their direction. It is obvious that mathematicians do not do this just for simple observation. To determine the accurately placement of that point, they study the second derivative of the function. Having in mind that the slope of the tangent straight line at that point is null (the red straight line shown in the graph), it is possible to accurately determine the existence of a point of inflection.

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