Mean value theorem and related statements

Here we introduce three core theorems of differential calculus.

Theorem (Rolle's theorem).
Let a < b be real numbers. Let f be a function continuous on the interval [a,b] and differentiable on (a,b).
If f (a) = f (b), then there exists some c in (a,b) satisfying

What is the meaning of this theorem? If the two ends of the graph are on the same level, then there must be a point where the tangent line is horizontal:

This looks like something that should be naturally true. To understand this theorem better we look at the importance of its main assumptions. There are several. Some assumptions are about the function: That it is continuous, that is has derivative, that its ends are on the same level. The following picture shows that violation of every single one of these three makes the conclusion invalid, that is, for those three examples we are not able to find a horizontal tangent line.

We also have assumptions about the set. A closed interval is actually a bounded, closed and connected set. Two of these are so basic that we cannot skip them without changing the setting so much that the statement becomes meaningless. When a set is not bounded, then it does not have both endpoints finite, so we cannot compare at which level the two ends of the graph are. Similarly we might lose endpoints for sets that are not closed. Thus the only really interesting condition about the set is its connectedness. In the following picture we show that if we violate it, we again cannot rely on having a horizontal tangent line. Note that the function in the picture is continuous on the indicated closed and bounded set M and differentiable on its interior, obviously both ends are at the same level.

The pictures show that if we skip even one of the assumptions, then we cannot rely any more on the existence of a horizontal tangent line. Of course, horizontal tangent lines can appear also for functions without these assumptions, for instance in this picture the function does not satisfy perhaps a single one and sill has a horizontal tangent line:

but it is just a question of luck, whereas we are interested in situations where we have it guaranteed. One more remark: Differentiability of f on (a,b) already implies that this function is continuous there. The second assumption (on continuity) thus actually brings only two new things, namely on-sided continuity at endpoints. That we do not get from differentiability.

If we take the picture that explained the Rolle theorem and tilt it, it seems that everything should still work. And so it does, it is just a matter of expressing things properly.

Theorem (Mean value theorem, Lagrange's theorem).
Let a < b be real numbers. Let f be a function continuous on the interval [a,b] and differentiable on (a,b). There exists some c in (a,b) satisfying

What is the meaning of this theorem? If we connect the two endpoints of the graph of f with a straight line, then its slope is exactly the fraction on the right. The theorem says that there must be a point where the tangent line has the same slope. That is, if we connect the two endpoints of the graph by a straight line, then there must be a tangent line that is parallel to this connecting line.

This theorem is based on Rolle's theorem, so the analysis of assumptions can be repeated here and similar pictures show that they are all needed. To put another way, the Rolle theorem is no just a special case of MVT, so if we drop an assumption and Rolle's theorem fails, it would violate also MVT.

This theorem has an interesting interpretation. Assume that f is a measurement of displacement (cf. for instance here or this note). Then the ratio on the right is actually the average velocity over the given interval, while the expression on the left is the instantaneous velocity at a certain time. This says that if we move in a reasonable manner (smoothly enough to have a derivative), then our average velocity must have been attained at some moment.

We conclude this section with a generalization of MVT.

Theorem (Cauchy's theorem).
Let a < b be real numbers. Let f and g be functions continuous on the interval [a,b] and differentiable on (a,b).
If g(a) ≠ g(b), then there exists some c in (a,b) satisfying

Now this theorem is a tough one to explain, since it cannot be really shown well in a picture. One possible way to approach this theorem is as follows. We went from Rolle's theorem to MVT by tilting the picture in Rolle. To get from MVT to Cauchy's theorem we take the variable x and transform it into a new variable using the formula g(x). If this transformation is simple (like stretching or moving the graph left/right), then it seems reasonable that the conclusion about parallel lines should still work, but this theorem also allows for more complicated transformations.

Note that if we take the Cauchy theorem and use it with the function g(x) = x, we get the Mean value theorem. So Cauchy's theorem is naturally the most powerful of the three, but it is so abstract that seeing it used in concrete situations is extremely rare. From practical point of view, MVT is the king. However, without Cauchy's theorem we would not get l'Hospital's rule in the next section, and imagine having to do limits without l'H. I don't think Cauchy's theorem needs any more credentials than this.


Derivative and limit
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