# Inflection, Point of

## Inflection, Point of

The point *M* of a plane curve is a point of inflection for the curve if the curve has a unique tangent at *M* and if in a sufficiently small neighborhood of *M* the curve is contained within one pair of the vertical angles formed by the tangent and normal (see Figure 1). The point (0, 0) of the curve *y* = x^{3} is an example of a point of inflection.

Suppose a curve is given by the equation *y = f(x*), where *f(x*) has the continuous second derivative *f” (x*). If the point with coordinates [*x*_{0}, *f* (*x*_{0})] is a point of inflection, then *f” (x*) = 0; thus, a curve has zero curvature at a point of inflection. For a point to be a point of inflection it is necessary but not sufficient that *f” (x*) = 0. For example, that equality is satisfied by the curve *y* = *x*^{4} at the point (0, 0), although this point is not a point of inflection. A complete investigation of whether a given point of a curve is a point of inflection requires the use of higher order derivatives (if such derivatives exist) or other supplementary evidence.