[Math] Why does “convex function” mean “concave *up*”

Question

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$,
$$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$
Equivalently, a line segment between two points on the graph lies above the graph, the region above the graph is convex, etc. I want to know why the word "convex" goes with the inequality in this direction, and how I can remember it. Every reason I have heard makes just as much sense applied to the opposite inequality ("concave down").

Answer 1

Not sure why convex is defined that way, but one way to remember is that the derivative is monotonically increasing for some convex functions.

Or maybe just remember that $e^x$ is conv$e^x$. (I just thought of this one!)