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Here is the definition for each kind of function https://i.stack.imgur.com/dMWb1.jpg And here is the definition for the words "concave and convex" in dictionary, with convex means curving out and concave means curving in. https://i.stack.imgur.com/99SkU.jpg

In my logic, the convex function should be called concave func, and concave the opposite, because from the graph, I see a rumble strip bar, going up a little bit and down, like a concave function, but I think the rumble strip bar is described as "convex", here it's a little difficult for you to see my logic about it, but I don't understand why we call those function like that, not the opposite. Sorry because i'm not in an English speaker country. Thanks for your explaination!

Feliks
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  • Otherwise the mnemonic “exp is convex” wouldn't work. ;-) – Hans Lundmark Oct 04 '16 at 18:37
  • I dont really understand your idea, can you say more? – Feliks Oct 05 '16 at 04:20
  • "Exp" (the exponential function, $f(x)=\exp x=e^x$) rhymes with "convex" but not with "concave". – Hans Lundmark Oct 05 '16 at 05:33
  • I agree, I just want to know why we call that not the opposite, because in my natural thought, I think it's the opposite (you can see the both images to see my idea clearly). Mitchell Spector explains that it's because the relationship between the graph and the horizontal line, not the x axis – Feliks Oct 05 '16 at 05:46
  • Notice the smiley! It was a joke... – Hans Lundmark Oct 05 '16 at 05:48
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    Anyway, if you look at the region above the graph of a convex function, it has a shape which is convex in the everyday sense of the word. The choice of above/below here is clearly quite arbitrary, but someone made the choice and it stuck. – Hans Lundmark Oct 05 '16 at 07:57
  • Related: https://math.stackexchange.com/questions/3399/why-does-convex-function-mean-concave-up – Peter O. Jan 10 '22 at 04:15

1 Answers1

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I don't know the historical origins of the mathematical usage, but I have always thought it was backward also, and I'm a native speaker of English. The terms in mathematics are relative not to the x-axis, but to a horizontal line above the curve (at $y=\infty$ if you like).

To avoid confusion, I think it's often better to use the terms "concave up" and "concave down", which are quite clear ("convex" = "concave up", and "concave" = "concave down").

Mitchell Spector
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  • So is the relationship between horizontal line and the graph the main and official reason for the names? My bottom line is to find a true reason to explain to people in my school and suggest a change in maths system. In my country, the first translator used the words for his logic about the graph, so now, we have the definition opposite to the world! :( – Feliks Oct 05 '16 at 04:18
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    I don't know the etymology, but the names are well entrenched and aren't going to be changing. There's no "official" reason, because there's no official body to rule on such things (either for the English language or specifically for mathematical usage in English). These definitions must have simply been adopted by somebody originally, and the usage caught on. – Mitchell Spector Oct 05 '16 at 04:54