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I always struggle to remember when a function is convex and concave:

enter image description here

Do you have a particular trick to help you remember this?

My trick is based on the Spanish phrase "No cabe", pronounced nô ˈka.βe, which sound just like "concave". "No cabe" means it does not fit. Thus, whilst you can put something into a convex function (e.g. think of a bowl), you cannot put something into a concave function. Hence the relation.

I am curious on what other, perhaps more efficient methods people use.

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    $\begingroup$ I personally prefer "convex exponential function" and conclude that the logarithm is concave. $\endgroup$ Commented Jul 19, 2017 at 17:29
  • $\begingroup$ If you know the standard English meaning of the words convex and concave, you can remember that for a convex function it is the epigraph that is convex, and for a concave function it is the epigraph that is concave. Failing that, the "cave" mnemonic mentioned by @SeanRoberson seems unforgettable. $\endgroup$
    – littleO
    Commented Jul 19, 2017 at 17:45
  • $\begingroup$ In Swedish, which probably luchonacho doesn't know, one can connect convex with växande, meaning growing. Here, vex and väx sound the same. A convex function doesn't have to be growing, but if it's differentiable, then the derivative is growing. $\endgroup$
    – md2perpe
    Commented Jul 19, 2017 at 20:17
  • $\begingroup$ @md2perpe: And "konkav" matches "avtagande" (=decreasing) too! :-) $\endgroup$ Commented Jul 20, 2017 at 9:03
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    $\begingroup$ Doubling down on the first comment. "$e$ to the $x$ is convex" (which even sort of rhymes) is how I remember it. Just remembering one example of a convex function is enough. $\endgroup$ Commented Feb 17, 2021 at 15:57

7 Answers 7

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I think it just depends on how you learn.

When I took calculus, we didn't use "concave" and "convex" - rather, we (and the AP exam) used "concave up" and "concave down." I still use these as a grad student.

One can also remember that concave functions look like the opening of a cave.

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  • $\begingroup$ Why not use convex up and convex down, though? $\endgroup$
    – k.stm
    Commented Jul 19, 2017 at 17:35
  • $\begingroup$ @k.stm Who knows. I just went with it. $\endgroup$ Commented Jul 19, 2017 at 17:37
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    $\begingroup$ To this I might add "concave up, like a cup" and "concave down, like a frown" if you have more difficulty remembering. $\endgroup$ Commented Jul 19, 2017 at 18:20
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    $\begingroup$ @BobKrueger I did something similar when dealing with curve sketching and derivatives - concave up, smiley face, $f''(x) > 0$. Concave down, frowny face, $f''(x) < 0$. $\endgroup$ Commented Jul 19, 2017 at 18:21
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In English, there is a great trick from David MacKay's recommended book on information theory: Just keep pronouncing the word "convex" as "convec-smile" and "concave" as "conca-frown" and the direction of the mouth of the corresponding smiley will tell you what the graph of the function looks like.

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Well, I think of it as they both are in a smiling contest, concave smiles with the v and convex with the v and x. However, the fact that the function convex smiles more makes the concave sad.

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conVex - V looks like the convex function :)

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    $\begingroup$ But concaVe.... $\endgroup$
    – F.A.
    Commented Apr 16, 2018 at 17:46
  • $\begingroup$ @F.A. But Co⋂cave... ;) $\endgroup$
    – The Quark
    Commented May 18, 2023 at 15:21
  • $\begingroup$ @TheQuark There is a "n" also in "convex" $\endgroup$ Commented Jun 3, 2023 at 17:05
  • $\begingroup$ @robertspierre Yes, thus my actual (different) answer to the question here $\endgroup$
    – The Quark
    Commented Jun 24, 2023 at 8:27
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you just need to remember 1 of them, conCAVE, it is already in the word it is a cave hence the shape :-)

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  • $\begingroup$ Duplicate of @Sean Roberson answer $\endgroup$ Commented Jun 3, 2023 at 17:04
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For consistency and coherence between curve shapes in a $(x,y)$ plane and the shape of the surface of lenses, it is better to relate the terms with the sign of the curvature $\partial^2 y/\partial x^2$:

  • concave = negative curvature: $\partial^2 y/\partial x^2 < 0$ (thus: downward curve or hollow surface)
  • convex = positive curvature: $\partial^2 y/\partial x^2 > 0$ (thus: upward curve or domed surface)

concAve <-> negAtive

conveX <-> $+$ (rotated X)

And when you look at a surface, the equivalent of the y-axis is the axis of your gaze, oriented away from you.

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I was taught in my language that concave = the one I can't pour coffee ("káva") in. Anyway it doesn't help that with lenses and mirrors this is the other way around.

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