What does it mean for a graph to be concave up or down over an interval?
Not an easy question. I have seen a book where the idea was defined solely with a picture! Now pictures are good, and you should certainly use pictures to give your students a good understanding of concavity, but a definition needs a bit more. But there seems to be no general agreement on a definition.
- My favorite definition is that a function is concave up (down) on an interval where any (every, all) line tangent to the graph in the interval lies below (above) the graph in the interval (except at the point of tangency).
- One of the most common definitions is that if the second derivative is positive (negative) on an interval, the graph is concave up (down) on the interval.
- Another approach is to connect any (every, all) pairs of points of the function in the interval. If all these segments (except their endpoints) lie above (below) the graph then the graph is concave up (down).
- Yet another is that a function is concave up (down) where the first derivative is increasing (decreasing). This is not quite the same as saying a function is concave up (down) where the first derivative is positive (negative), because of the question of including or excluding the endpoints (see the post of November 2, 2012), but this too could be a definition.
The second bullet above is used to find where the graph is concave up or down. The first idea is still true and it is important to know when trying to determine whether a tangent line approximation is larger or smaller than the actual value. If the tangent line between the point of tangency and the approximated point is below the curve (that is, the curve is concave up) the approximation is an underestimate (smaller) than the actual value; if above, then an overestimate.)
A graph is concave up where its second derivative is positive and concave down where its second derivative is negative. Thus, the concavity changes where the second derivative is zero or undefined. Such a point is called a point of inflection.
The procedure for finding a point of inflection is similar to the one for finding local extreme values: (1) find where the second derivative is zero or undefined, (2) determine that the sign of the second derivative changes, and then (3) identify the point of inflection.
Finally, a pet peeve:
Back in Algebra I students learn all about the parabola y = ax2. They learn that when a is positive the graph “opens upward” or “holds water” or is a “smiley face.” Why? Why not use the correct terms – concave up and concave down – right there, right from the start?
Next: Analyzing the graph of the derivative.
Revised October 7, 2014
What about the intervals? Are they open or closed? A cubic, for example is concave up (0, infinity) or [0, infinity)?
As I mentioned and as Andrew Ross wrote, it depends on the definition you or your textbook use. If you say a function is concave up where the second derivative is strictly positive (bullet 2) then for your example the interval is open: (0, infinity). If you instead use a function is concave up where the first derivative is increasing (new bullet 4) then half-open: [0, infinity).
(On including the endpoints for increasing functions see my post of November 2, 2012 and also here.)
Your 3rd bullet point is the best _definition_ of concave up or down, I think, because it doesn’t require the function to be differentiable. A common example: piecewise-linear functions like income tax $ as a function of income. The slope increases as you move to the right (if it’s a progressive tax system). At the breakpoints, the function isn’t differentiable/has no unique tangent line, but we’d still want to call the function concave-up. This is the common way to define CU and CD in “operations research”, a subfield of applied math that includes a lot of optimization (usually multivariable).
However, I would change “above” the graph to “on or above”, that is, allow >= instead of just > for CU, and use the term “strictly CU” for the > case. This means that purely linear functions are simultaneously CU and CD (though they are not strictly CU or strictly CD), even though I just read in the Hughes-Hallett book that they are neither CU nor CD. We want to be inclusive here because some important theorems are still true for linear or piecewise linear functions: any local minimum of a concave-up function on a convex set is also a global minimum, and the maximum of a concave-up function on a convex set occurs at an extreme point.
It is common in operations research to use “convex” for “concave up” and “concave” for “concave down”. I agree that the choice seems arbitrary (like the sign of charge on an electron); CU and CD are much more descriptive. But if you want to understand what many people are writing about optimization, you need to know the vocabulary they’re using. Perhaps not much of an issue at the Calc I level, though.
Just found this resource and I can’t wait to explore. On the comment of concave up and concave down, I learned those terms as well back in the day but I was watching a video from MIT (those iTunes University ones) and the instructor mentioned concave and convex and mentioned hopefully being rid of the terms concave up and concave down. Any thoughts migrating to those terms? I personally like up and down since it’s one less thing to remember but who knows…
I agree with you. One less term to memorize. Outside of dealing with functions convex may be more useful, but there is no need for an extra term here. Besides, which way is convex? Does it replace concave up or concave down? And why that way and not the other way?