If an increasing function is differentiable, it has a positive first derivative.

If a convex function is twice differentiable, it has a positive second derivative.

If a ______ function is thrice differentiable, it has a positive third derivative.

Is there such a word that describes this general notion without relying on derivatives?

Someone suggested to me there is such a word (or perhaps a word corresponding to a negative third derivative), and it begins with the letter B, but that isn’t much help.

(Yes I understand if it’s differentiable one time, then the derivative is convex. That’s not what I’m looking for. And I’m not worried about strict v. weak interpretation, the derivative could be nonnegative.)

A positive third derivative tells you that the convexity/concavity is increasing/decreasing, not sure there’s a word for a function where that is strictly true.

I suppose you could say something like “the curve is tightening”. Though then you get into the distinction between the second derivative and the curvature.

The statement as given is false if “positive” means strictly > 0. A twice-differentiable function on an interval is convex if and only if its 2nd derivative is non-negative, and strictly convex if the 2nd derivative is positive, but strict convexity does not imply the second derivative is everywhere positive.

I know “jerk” is the third derivative for position. And before anyone suggests it I know “prudence” is the third derivative for utility. But note that both of those actually depend on the third derivative existing.

I’m aware of this. I said in the OP I was not worried about strictness.

Further information: It turns out the “B” suggestion was a joke because in bond pricing we have “Duration” and “Convexity” so the next derivative should begin with a B.

In any case, it appears there is no generally accepted word, so I’ll have to be satisfied with simply describing it.

Thanks (particularly to Chronos who made a suggestion)

This statement makes me worried. Now, I don’t know the specific definition of “utility”, and , frankly, I don’t care much, but the mere fact that there are people (hundreds or thousands maybe?) who actually need a special word for the third derivative of this concept (with respect to the same variable thrice?) makes me feel all the more alienated from the ruling class…

For a function with a strictly positive third derivative the curve would “tighten” over intervals where the function is convex, but “loosen” over intervals where the function is concave.

Take f(x)=x[sup]3[/sup] where f’’’(x)=6 and so the function has a strictly positive third derivative. The function is concave over the interval (-∞,0), but is convex over the interval (0,∞).