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 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