A point on a graph y=f(x) at which the concavity changes. Thus, there is a point of inflexion at a if, for some δ>0, f″(x) exists and is positive in the open interval (a−δ, a) and f″(x) exists and is negative in the open interval (a, a+δ), or vice versa.
If the change (as x increases) is from concave up to concave down, the graph and its tangent at the point of inflexion look like one of the cases shown in the first row of the figure. If the change is from concave down to concave up, the graph and its tangent look like one of the cases shown in the second row. The middle diagram in each row shows a point of inflexion that is also a stationary point, since the tangent is horizontal.
If f″ is continuous at a, then for y=f(x) to have a point of inflexion at a it is necessary that f″(a)=0, and so this is the usual method of finding possible points of inflexion. However, the condition f″(a)=0 is not sufficient to ensure that there is a point of inflexion at a: it must be shown that f″(x) is positive to one side of a and negative to the other side. Thus, if f(x)=x4 then f″(0)=0; but y=x4 does not have a point of inflexion at 0 since f″(x) is positive to both sides of 0. Finally, there may be a point of inflexion at a point where f″(x) does not exist, for example at the origin on the curve y=x1/3, as shown in the figure.