In Calculus I students are taught how to find the points at which the graph of a function has zero curvature (that is, the points of inflection). The points of maximal curvature are usually not discussed. This post attempts to rectify this omission.

The (signed) curvature of is

We want to maximize the absolute value of , whether the function is positive or negative there (so both maxima and minima of can be of interest). The critical points of are the zeros of

So we are lead to consider a polynomial of the first three derivatives of , namely .

Begin with some simple examples:

has so the curvature of a parabola is maximal at its vertex. No surprise there.

has , indicating two symmetric points of maximal curvature, , pretty close to the point of inflection.

has . This has three real roots, but actually minimizes curvature (it vanishes there).

More generally, with positive integer yields indicating two points which tend to as grows.

The graph of a polynomial of degree can have at most points of zero curvature, because the second derivative vanishes at those. How many points of maximal curvature can it have? The degree of expression above is but it is not obvious whether all of its roots can be real and distinct, and also be the maxima of (unlike for ). For we do get point of maximal curvature. But for , there can be at most such points, not . Edwards and Gordon (Extreme curvature of polynomials, 2004) conjectured that the graph of a polynomial of degree **has at most points of maximal curvature**. This remains open despite several partial results: see the recent paper Extreme curvature of polynomials and level sets (2017).

A few more elementary functions:

has , so the curvature is maximal at . Did I expect the maximum of curvature to occur for a negative ? Not really.

has . The first factor is irrelevant: the points of maximum curvature of a sine wave are at its extrema, as one would guess.

has which is zero at… ahem. The expression factors as

Writing we can get a cubic equation in , but it is not a nice one. Or we could do some trigonometry and reduce to the equation . Either way, a numerical solution is called for: (and for other periods).