The terms "concave up" and "concave down" seem to only appear in introductory calculus textbooks written for freshmen college students in USA. Elsewhere in mathematics, we don't use such terms.
I wonder where and when these terms were introduced and why. Especially since we have a perfectly good and much more entrenched word: Convex.
The notion of a curve being concave in some direction is not newer than the modern notion of a convex function. Here is how the terminology of concavity has developed over the centuries. We can start at least as early as L’Hôpital’s Analyse des Infiniment Petits in 1696.
Lorsqu’une ligne courbe AFK est en partie concave et en partie convexe vers une ligne droite AB ou vers un point fixe B; le point F qui sépare la partie concave de la convexe, et qui par conséquent est la fin de l’une et la commencement de l’autre, est appellé point d’infléxion…
Early on, authors wrote that curves were concave towards or convex towards a line or point. In English for example, this is from Introduction to the Doctrine of Fluxions by John Rowe in 1751.
When a Curve from being Concave becomes Convex towards its Axis, or from being Convex becomes Concave, that Point where the Change is made, or that which separates the Convex from the Concave Part, is called the Point of Inflection…
At some point, English authors started using the terminology of curves being concave or convex upwards or downwards.
For example in John Hind’s 1831 The Principles of the Differential Calculus, he first uses the terms convex towards the x-axis and concave towards the x-axis in his section the direction of curvature. But when he discusses the example of the right strophoid, he writes that “the symmetrical branches are respectively concave upwards and downwards.”
In Edward Albert Bowser’s 1885 An Elementary Treatise on the Differential and Integral Calculus, he begins the section on direction of curvature by discussing a curve being concave or convex towards the x-axis, but then immediately begins discussing the notions of concave and convex downwards.
If a curve is concave downwards… the derivative of $\frac{dy}{dx}$ or $\frac{d^2y}{dx^2}$ is negative.
In the same way if the curve is convex downwards… the derivative of $\frac{dy}{dx}$ or $\frac{d^2y}{dx^2}$ is positive.
In The Elements of the Differential and Integral Calculus by Donald Francis Campbell in 1904, Campbell introduces the terms “concave upwards” and “concave downwards” in relation to slopes of tangent lines increasing or decreasing. He never uses the word convex to describe a curve. Within a few years, many textbooks were abbreviating the terms to “concave up” and “concave down.”
Typical calculus textbooks used in the United States have not deeply changed since the early to mid 1900s. The presentation has changed, but the substance is largely the same.
I am not as familiar with textbooks used at a similar level in other countries. From my limited knowledge, I have a hypothesis that I currently don’t have time to research. I believe the modern study of convexity and convex functions emerged more recently and at a relatively more advanced level than the notion of curves being concave in some direction. This idea made its way into analysis textbooks both in the U.S. and elsewhere, but U.S. calculus textbooks calcified before this.
Perhaps the origin can be traced to Newton's Bucket. There is a good Wikipedia article, and the article Newton’s Rotating Water Bucket: A Simple Model by Mungan and Lipscome is readable.
Newton writes about a bucket that holds water, suspended by a cord. The cord is twisted and then the bucket is released so that it can spin. The profile of the surface of the spinning mass of water develops into a parabolic shape. Newton of course wrote in Latin, but translations use the word concave to describe this shape. In the nomenclature of modern calculus textbooks, this shape would be classified as "concave up."
