inflexion point

In examining the graphs of differentiable real functions, it may be useful to state the intervals where the function is convex and the ones where it is concave.



Since the sine function is $ 2\pi$ -periodic, the sinusoid possesses infinitely many inflexion points. Indeed, ; for ; , . Non-nullity of the third derivative at these critical points assures us the existence of those inflexion points.

Remarks

1. For finding the inflexion points of the graph of it does not suffice to find the roots of the equation , since the sign of does not necessarily change as one passes such a root. If the second derivative maintains its sign when one of its zeros is passed, we can speak of a plain point (?) of the graph. E.g. the origin is a plain point of the graph of .

2. Recalling that the curvature for a plane curve is given by

we can say that the inflexion points are the points of the curve where the curvature changes its sign and where the curvature equals zero.

3. If an inflexion point satisfies the additional condition , the point is said to be a stationary inflexion point or a saddle-point, while in the case it is a non-stationary inflexion point.



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by pahio.