How to remember which function is concave and which one is convex?

Question

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

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

conVex - V looks like the convex function :)

Answer 5

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.

Answer 6

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.

Answer 7

you just need to remember 1 of them, conCAVE, it is already in the word it is a cave hence the shape :-)