Was there a more intuitive early proof of the generalized mean value theorem?

Question

I am interested in the early proofs of the theorem. It is often called Cauchy mean value theorem, so perhaps Cauchy proved it first. In all the proofs that I have seen we construct a contrived function, applying Rolle's theorem to which works out just right. It seems like a miracle that one would come up with such a function. Perhaps the older proofs were more intuitive.

Answer 1

Cauchy was indeed the first, although his version was weaker than the modern one. An intuitive geometric interpretation of the theorem, along with a proof motivated by it, is due to Bonnet. A very well sourced and illustrated account of history of the mean value theorem is A brief history of the mean value theorem by Besenyei, which describes many historically given proofs of it and its generalization, along with controversies surrounding some of them.

Cauchy's proof appeared in Resume des Lecons sur le Calcul Infinitesimal (1823). He does not use Rolle's theorem, but he also does not prove the modern form of it involving derivatives at an intermediate point. In Besenyei's reconstruction, we assume $g′ > 0$, and denote $A := \min_{[a,b]} f ′/g′$, and $B := \max_{[a,b]} f ′/g′$. Then $f' − Ag'\geq0$ and $Bg' − f'\geq0$, and hence $f − Ag$ and $Bg − f$ are monotone increasing. Therefore, $$ f (b)−f (a)−A(g(b)−g(a)) ≥ 0\ \textrm{and}\ B(g(b)−g(a))−(f (b)−f (a)) ≥ 0. $$ These are easily manipulated into the double inequality $$ A ≤ \frac{f (b) − f (a)}{g(b) − g(a)} ≤ B, $$ which is Cauchy's version of the theorem.

The modern version appeared in Serret's Cours de Calcul Infinitesimal (1868). The auxiliary function Serret introduced is directly inspired by Cauchy's proof, it is $\varphi(x) = (f(x) − Ag(x)) − (f(a) − Ag(a))$, but Rolle's theorem is not used.

Bonnet gave an intuitive geometric interpretation of the theorem, by treating $g(t),f(t)$ as parametric equations of a plane curve. Then the theorem amounts to saying that the tangent to the curve segment at some point is parallel to the chord connecting its endpoints, $\big(g(a),f(a)\big)$ and $\big(g(b),f(b)\big)$. The slope of the chord is $\frac{f(b)−f(a)}{g(b)−g(a)}$, which suggests introducing an auxiliary function $$ h(t) = f (t) − f (a) − \frac{f(b)−f(a)}{g(b)−g(a)}(g(t) − g(a)). $$ Geometrically, this is the vertical distance between $(g(t),f(t))$ and the point on the chord right above or below it. The generalized mean value theorem follows by applying Rolle's theorem to $h(t)$.