What are inflection points?

The inflection point of a function is the point where the function changes its curvature. So if is was curved to the left first, it would be curved to the right after the inflection point. Intuitively, the inflection point is found where the slope of the function reaches its highest value. For example, when you are walking up a hill and its slope becomes steeper and steeper, then you reached its inflection point when the slope becomes less steep (the hill starts to curve down). Mathematically, the inflection is found where the first derivative has an extremum (minimum or maximum).

We know that to find the extremum of a function equation the following conditions must be met:

(1)   equation

Because we want to find the extrema of the derivative equation and not the function equation, the following two conditions must be met for an inflection point:

(2)   equation

Further, if equation, then the function changes its curvature from right to left and vice-versa if equation.

Example

Consider the following function and its derivatives:

(3)   equation

We now find the roots of the second derivative equation by setting it zero and solving for equation:

(4)   equation

We have a potential inflection point at equation. To confirm this, we evaluate the third derivative at this point:

    equation

Because equation, we have an inflection point and because it is greater than zero, the curvature changes from right to left. The following graph illustrates this. The function is shown in red and it is obvious that it is turning from right to left. The first derivative (purple) has a minimum at this point. The second derivative (orange) has a root at this point. The third derivative (black) is positive at this point (in fact it is constant and positive everywhere).

inflection