**The inflection point of a mathematical function is that point at which the graph that represents it changes its concavity. That is, it goes from being concave to being convex, or vice versa.**

The inflection point, in other words, is that moment when the function changes trend.

To get an idea, let’s start by looking at it in a graphic representation, roughly:

It should be noted that a function can have more than one inflection point, or not have them at all. For example, a line does not have an inflection point.

Let’s see, in the following graph, an example of a function with more than one inflection point:

Also, in mathematical terms, the inflection point is calculated by setting the second derivative of the function equal to zero. Thus, we solve for the root (or roots) of that equation and we will call it Xi.

Then, we replace Xi in the third derivative of the function. If the result is different from zero, we are facing an inflection point.

However, if the result is zero, we must replace in the successive derivatives, until the value of this derivative, be it the third, fourth or fifth, is different from 0. If the derivative is odd, it is an inflection point , but if it is even no.

## Turning point example

Next, let’s look at an example.

Suppose we have the following function:

y = 2x ^{4} + 5x ^{3} + 9x + 14

y ‘= 8x ^{3} + 15x ^{2} +9

y »= 24x ^{2} + 30x = 0

24x = -30

Xi = -1.25

Then we replace Xi in the third derivative:

y »’= 48x

y »’= 48x-1.25 = -60

As the result is different from zero, we find ourselves in front of an inflection point that would be when x is equal to -1.25 and y is equal to -2.1328, as we illustrate in the following graph.

In this it is observed that the function has an inflection point:

Now, let’s look at another example:

y = x ^{4} -54x ^{2}

y ‘= 4x ^{3} -108x

y »= 12x ^{2} -108 = 0

x ^{2} = 9

Xi = 3 and -3

Then, we replace the two roots found in the third derivative:

y »’= 24x

y »’= 24 × 3 = 72

y »’= 24x-3 = -72

Since the result is nonzero, we have two inflection points at (3,567) and (-3,567).

To complement the information, we invite you to visit the inflection article, where we cover this concept more generally: