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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. $\endgroup$ – Mundron Schmidt Jul 19 '17 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 Jul 19 '17 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 Jul 19 '17 at 20:17
  • $\begingroup$ @md2perpe: And "konkav" matches "avtagande" (=decreasing) too! :-) $\endgroup$ – Hans Lundmark Jul 20 '17 at 9:03
  • $\begingroup$ Det är så sant så! $\endgroup$ – md2perpe Jul 20 '17 at 22:51
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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 Jul 19 '17 at 17:35
  • $\begingroup$ @k.stm Who knows. I just went with it. $\endgroup$ – Sean Roberson Jul 19 '17 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. $\endgroup$ – Bob Krueger Jul 19 '17 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, $f''(x) < 0$. $\endgroup$ – Sean Roberson Jul 19 '17 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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conVex - V looks like the convex function :)

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    $\begingroup$ But concaVe.... $\endgroup$ – F.A. Apr 16 '18 at 17:46

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