Concavity

In this section we will assume that the function that we investigate is "nice" in the sense that its domain can be split into subintervals in such a way that on each of them the function has some concavity and is twice differentiable. Of course, not every function has this property, for instance the Dirichlet function (in Functions - Theory - Elementary functions) does not have a single interval in its domain on which it would be concave up or down. Fortunately, functions we meet almost always fall into the "nice" category.

The main tool for identifying intervals of concavity is the theorems from Derivative and concavity in Theory - MVT. From them we can deduce the following:

Fact.
Let f be a function. Assume that for some a < b < c this function is concave up on (a,b) and concave down on (b,c), or concave down on (a,b) and concave up on (b,c). Then either f ′′(b) = 0 or f does not have the second derivative at b.

This fact is an implication, so not every point with f ′′ zero or non-existent separates intervals of different concavity, but such points are natural candidates. They do not have a special name.

The first step in investigating concavity of a function f : We start with the intervals that constitute the domain of f. Then we find all points with f ′′ zero or non-existent (it is not necessary to precisely investigate the existence of the second derivative, it is enough to include all points where this derivative is suspect). These will further split the intervals of the domain into subintervals. On each of the resulting intervals the function satisfies some concavity property.

To find out what kind of concavity we have on these intervals we use this theorem:

Theorem.
Let f be a function continuous on an interval I and twice differentiable on its interior Int(I ).
If f ′′ > 0 on Int(I ), then f is concave up on I.
If f ′′ < 0 on Int(I ), then f is concave down on I.

The second step in investigating concavity of a function f : For each interval from the first step we find out what sign the second derivative has inside this interval. Since the sign must be the same on such an interval, the easiest way to determine the signs is to pick some point from inside the interval and put it into f ′′. For an example look below.

Question: When do we put open and when closed intervals in the answer?
Answer: We usually include endpoints if the functions is continuous at such a point from appropriate side (for a right endpoint we need continuity from the left and vice versa).

Points of inflection are, by definition, points where the function exists and changes from one concavity to the other. Thus we find them easily by looking at concavity intervals.

We refer to Concavity in Methods Survey - Graphing for a practical overview.

Example: Investigate concavity of the function f (x) = x4 − 4x3.

Solution: The domain is the whole real line, so there is one starting interval, namely (−∞,∞).
We find the second derivative: f ′′(x) = 12x2 − 24x = 12x(x − 2).
Dividing points: f ′′(x) = 0 yields x = 0 and x = 2; there are no points where f ′′ does not exists.
Thus we have three intervals of concavity: (−∞,0], [0,2], and [2,∞), we used closed intervals where f is continuous.

We substitute x = −1 into f ′′ to see that f ′′ > 0 on (−∞,0]. Substituting, say, x = 1 into f ′′ we see that f ′′ < 0 on [0,2]. Finally, we try x = 3 to find out that f ′′ > 0 on [2,∞).

The conclusion is that f is concave up on (−∞,0], concave down on [0,2] and concave up on [2,∞). There are two points of inflection, f (0) = 0 and f (2) = −16. By the way, the function looks like this:

This procedure can be streamlined using a table, see Concavity in Methods Survey - Graphing.


Asymptotes
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