How to Find Inflection Points of a Function - Definition, Formula, Example

Definition:

The point in a curve, where the curvature sign changes is called as the inflection point. It is also called as inflection point.

Method :

To find inflection point we need to integrate quadratic expression twice. Here are the steps to find out the inflection point.
Consider the quadratic expression f(x) = 2x³ + 2x² + 1

Step: 1 (First Step of Integration) f(x) = 2x³ + 2x² + 1 By integrating 2x³ + 2x² + 1, we will get f'(x) = 6x² + 4x

Step: 2 (Second Step of Integration) By integrating f'(x) = 6x² + 4x, we will get f''(x) = 12x + 4

Step: 3 Equalize the integrated expression to 0 to find the value of x. Hence, f''(x) = 12x + 4 = 0 12x = -4 x = -1/3 or -0.3333

Step: 4 Substitute the x value to the given (input) expression f(x) = 2x³ + 2x² + 1 f(x) = 2(-1/3)³ + 2(-1/3)² + 1 f(x) = 1.1481

Therefore the inflection point for the given quadratic expression f(x) = 2x³ + 2x² + 1 is (1.1481 at x = -0.3333)