Editing Inflection point (section)

==Categorization of points of inflection==
[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''<sup>4</sup> – ''x''}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]

Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.
* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''
* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''

A stationary point of inflection is not a [[local extremum]]. More generally, in the context of [[functions of several real variables]], a stationary point that is not a local extremum is called a [[saddle point#Mathematical discussion|saddle point]].

An example of a stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. The tangent is the {{mvar|x}}-axis, which cuts the graph at this point.

An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.