Average of a set of critical points

Question

Consider the following functions: $f(x) = \sin(x)$and $g(x) =x^3 + 1$. Let set $A$ include only the critical point(s) of $f$ across all positive reals and set $B$ include only the critical points of $g$ across all reals. Let $A\cup B$ denote the union of sets $A$ and $B$. Find the average of $A\cup B$. I asked a question similar to this but forgot to add the word positive, which greatly affects the average.

Edit: I am defining an average as an arithmetic mean of numbers. Whether this is applicable or not to an infinite set, I do not know.So the mean of an infinite set seems more like a measure, or so to speak, Infinite averages

Answer 1

To make any sense out of the idea of the "average" of an infinite set of numbers, your numbers must at least be

1. countably infinite
2. bounded

For example, the positive critical points of $f(x)=\sin(x)$, by which I will assume you mean the positive solutions of $\dfrac{d}{dx}\sin(x)=\cos(x)=0$, which consists of all odd multiples of $\dfrac{\pi}{2}$.

$$\frac{\pi}{2},\frac{3\pi}{2},\frac{5\pi}{2},\frac{7\pi}{2}.\cdots,\frac{(2n-1)\pi}{2}$$

The average of the first $n$ of these terms would just be $\frac{\pi}{2}$ times the average of the first $n$ odd integers.

But it is known that the sum of the first $n$ odd integers is $n^2$. For example

1. The sum of just the first odd integer is $1=1^2$
2. The sum of the first two odd integers is $1+3=2^2$
3. The sum of the first three odd integers is $1+3+5=3^2$

and so forth.

Thus the average of the first $n$ odd integers equals their sum $n^2$ divided by $n$. This means that the average of the first $n$ odd integers equals $n$.

So the average of the first $n$ odd multiples of $\dfrac{\pi}{2}$, the first $n$ positive critical points of $f(x)=\sin(x)$ is $A_n=\dfrac{n\pi}{2}$

For example, the average of the first $2,000,000$ critical points would be $1,000,000\pi$.

So you can never find the average of all of them since it increases without bound.

Perhaps you should ask whether you can find the average of the positive critical points of $f(x)=\sin\left(\dfrac{1}{x}\right)$ which is a bounded set whose graph is below:

One final thing to think about: Does a countable set really have to be bounded to have an average?

For example, the set of partial sums of the harmonic series is an unbounded set. Does this set have an average? Does

$$\dfrac{1}{n}\sum_{k=1}^{n}\frac{1}{k}$$

converge?