Inflection Points


How to Determine an Inflection Point

An inflection point is a point that when substituted in the second derivative, equals to 0.


Worked Example:

In this example, we will try to find the inflection point of the equation y=x^3. ​


Step 1: Find the Double Derivative (Differentiate Two Times)

Horizontal Point of Inflection

A horizontal point of inflection is a point with a gradient of 0 and has a positive or negative gradient on both sides. A horizontal point of inflection is also an inflection point as the double derivate will equal to 0 at its co-ordinates.


A horizontal point of inflection with both sides being positive is a positive horizontal point of inflection. A horizontal point of inflection with both sides being negative is a negative horizontal point of inflection.


We can use the sign test to prove that (0,0) is a horizontal point of inflection in the curve y=x^3.

Applications of Differentiation

Step 2: Make the Double Derive Equal 0 and Solve for X

Step 2: After finding x, use the equation to find y-co-ordinates of the inflection point.

You can also use the sign test to determine a horizontal point of inflection. 

Regular Inflection Points


Inflection points that are not horizontal points of inflection represent the largest gradient between two stationary points. It represents the point where the concavity changes, such as a transition from a minimum turning point to a maximum turning point.

At the inflection point, the gradient stops decreasing or increasing in value between two inflection points.

Below shows the graph of y=3x^3-2x and its inflection point of (0,0).

Worked Example of Determining (0,0) as the Inflection Point