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.)
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…