How To Find An Inflection Point

Inflection points identify where the concavity of a curve changes. This knowledge can be useful for determining the point at which a rate of change begins to slow or increase or can be used in chemistry for finding the equivalence point after titration. Finding the inflection point requires solving the second derivative for zero and evaluating the sign of that derivative around the point where it equals zero.

Take the second derivative of the equation of interest. Next, find all values where that second derivative equals zero or does not exist, such as where a denominator equals zero. These two steps identify all possible inflection points. To determine which of these points are actually inflection points, determine the sign of the second derivative on either side of the point. Second derivatives are positive when a curve is concave up and are negative when a curve is concave down. Therefore, when the second derivative is positive on one side of a point and negative on the other side, that point is an inflection point.