# General notion for positive third derivative **URL:** \ **Category:** Factual Questions\ **Created:** [April 19, 2018, 6:05pm UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723 "2018-04-19T18:05:10Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [April 19, 2018, 6:05pm UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/1 "2018-04-19T18:05:10Z")
Help me fill the blank in the sentence below. For functions of one variable: 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.) ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [April 19, 2018, 10:04pm UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/2 "2018-04-19T22:04:22Z")
f’ \> 0 –\> _increasing_ f’’ \> 0 –\> _accelerating_ f’‘’ \> 0 –\> _[jerking](https://en.wikipedia.org/wiki/Jerk_(physics))_ :eek: ---
**Author:** ![Ignotus](https://avatars.discourse-cdn.com/v4/letter/i/ecc23a/32.png) [@Ignotus](https://boards.straightdope.com/u/Ignotus)\ **Post date:** [April 19, 2018, 10:26pm UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/3 "2018-04-19T22:26:34Z")
And follow _jerk_ with _snap_, _crackle_ and _pop_! Seriously, are you trying to construct a jargon or to be understood by the great majority? ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [April 19, 2018, 11:08pm UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/4 "2018-04-19T23:08:40Z")
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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 19, 2018, 11:33pm UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/5 "2018-04-19T23:33:00Z")
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. ---
**Author:** ![Melbourne](https://avatars.discourse-cdn.com/v4/letter/m/b5e925/32.png) [@Melbourne](https://boards.straightdope.com/u/Melbourne)\ **Post date:** [April 20, 2018, 12:09am UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/6 "2018-04-20T00:09:29Z")
> [@OldGuy](#): > > . > > If a convex function is twice differentiable, it has a positive second derivative. ??? I know that’s not the question, but … ---
**Author:** ![Ignotus](https://avatars.discourse-cdn.com/v4/letter/i/ecc23a/32.png) [@Ignotus](https://boards.straightdope.com/u/Ignotus)\ **Post date:** [April 20, 2018, 12:15am UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/7 "2018-04-20T00:15:47Z")
What’s wrong with a phrase like _twice_ (or _thrice_) _increasing_? Meaning the second (or third) derivative is positive? ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 20, 2018, 12:26am UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/8 "2018-04-20T00:26:05Z")
> [@Melbourne](#): > > ??? > > I know that’s not the question, but … 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. ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [April 20, 2018, 12:45am UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/9 "2018-04-20T00:45:57Z")
> [@septimus](#): > > f’ \> 0 –\> _increasing_ > f’’ \> 0 –\> _accelerating_ > f’‘’ \> 0 –\> _[jerking](https://en.wikipedia.org/wiki/Jerk_(physics))_ > > :eek: 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. > [@DPRK](#): > > 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’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) ---
**Author:** ![Ignotus](https://avatars.discourse-cdn.com/v4/letter/i/ecc23a/32.png) [@Ignotus](https://boards.straightdope.com/u/Ignotus)\ **Post date:** [April 20, 2018, 1:03am UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/10 "2018-04-20T01:03:32Z")
> [@](#): > > And before anyone suggests it I know “prudence” is the third derivative for utility. 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… ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [April 20, 2018, 1:10am UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/11 "2018-04-20T01:10:27Z")
> [@Chronos](#): > > 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. 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,∞). ---
**Author:** ![glowacks](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/glowacks/32/5548_2.png) [@glowacks](https://boards.straightdope.com/u/glowacks)\ **Post date:** [April 20, 2018, 2:58am UTC](https://boards.straightdope.com/t/general-notion-for-positive-third-derivative/812723/12 "2018-04-20T02:58:49Z")
> [@Ignotus](#): > > 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… It’s apparently some jargon in the theory of risk aversion. See [Prudence - Wikipedia](https://en.wikipedia.org/wiki/Prudence#In_economics) and even more math at [Risk aversion - Wikipedia](https://en.wikipedia.org/wiki/Risk_aversion)