Inflection Point - Categorization of Points of Inflection

Categorization of Points of Inflection

Points of inflection can also be categorised according to whether f′(x) is zero or not zero.

• if f′(x) is zero, the point is a stationary point of inflection, also known as a saddle-point
• if f′(x) is not zero, the point is a non-stationary point of inflection

An example of a saddle point is the point (0,0) on the graph y = x3. The tangent is the x-axis, which cuts the graph at this point.

A non-stationary point of inflection can be visualised if the graph y = x3 is rotated slightly about the origin. The tangent at the origin still cuts the graph in two, but its gradient is non-zero.

Note that an inflection point is also called an ogee, although this term is sometimes applied to the entire curve which contains an inflection point.