Generalisation of Cauchy's mean value theorem I apologise in advance if this is an elementary question more fitted for Math Stack Exchange. The reason why I have decided to post here is that the question I am used to seeing on that site are not of the open-ended format of the one I am asking.
It is now the second time I have been studying Calculus (first self-taught, now in school) and we are going over the proof of Cauchy's mean value theorem (the precursor to l'Hopital's rule). I do understand the proof, and the intuitive explanation about parametrised curves in a plane, but I still think the statement of the theorem looks relatively obscure. Why are we considering a ratio and not something else? This led me to try and generalise, and this is where we get to my question.
I first tried finding a function $h: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that for any functions $f,g: D \subset \mathbb{R} \rightarrow \mathbb{R}$ satisfying Cauchy's mean value theorem's hypotheses, for any interval $[a,b] \subset D$, there exists $x \in [a,b]$ such that $h(f'(x), g'(x))=h(f(b-a), g(b-a))$. Beyond making a few tries and finding a few counterexamples, I realised this wasn't really in the spirit of a mean value theorem: we are trying to make an analogy, if we may use this term, between $f'(x)$ and $f(b)-f(a)$, while in both Lagrange and Cauchy's mean value theorems the analogy is made between $f'(x)$ and $\frac{f(b)-f(a)}{b-a}$. So I started looking for $h$ such that there exists $x$ such that
$$
h(f'(x), g'(x))= h\left(\frac{f(b)-f(a)}{b-a}, \frac{g(b)-g(a)}{b-a}\right).
$$
But this didn't really lead me anywhere.

The question I'm asking is precisely this: can we say anything more about functions $h(x,y) \neq \frac{x}{y}$ satisfying these statements? Suppose we simplify even further, and consider, for example, only the functions $h_{\alpha,\beta}(x,y) = x^{\alpha}y^{\beta}$. Can we maybe prove that only those with $\alpha = k, \beta = -k$ for some $k$ work? (in addition, clearly, to those with $\alpha\beta = 0$) Is this even interesting to investigate?

Thanks in advance for helping me. My knowledge doesn't really go far beyond Calculus and Linear Algebra (say, Spivak and Axler's books) but I will try to understand your replies.
 A: One may investigate such generalizations, but let's make a step backwards to your first question: Why are we considering a ratio? The reason is that those ratios are what we mainly care about, as we want to compare finite differences and differentials. And, as Dieudonné remarks in Foundations of Modern Analysuis, the point is not the existence of the point $x$, a result which is only true for scalar functions, and of which we usually can't say anything more than $x$ lies somewhere between $a$ and $b$. Rather, it is the inequality that follows from it, and which holds true even for vector valued curves: the "true" Mean Value Theorem is the inequality
$$ \|f(b)-f(a)\|\le(b-a)\sup_{a<x<b}\|f'(x)\|$$
for a differentiable curve $f:[a,b]\to E$ in a normed space $(E,\|\;\|)$.
The proof is a mathematical jem : take any $M>\sup_{a<x<b}\|f'(x)\|$. By a fist order expansion, no $t\in[a,b)$  can be a minimum of $\|f(t)-f(a)\|-Mt$. But by Weierstrass theorem, there is a minimum, so it is $b$, and the thesis follows.  (It's the same indirect method employed in the variational proof of the fundamental theorem of algebra, if you notice).
But why Lagrange MVThm is so important, that it could compete for the title of "fundamental theorem of differential calculus"?
Because the whole theory of differential calculus in Banach spaces is built on it. All these theorems are few lines consequences of it :

*

*For a differentiable function on a convex $\Omega$, $\|df\|_{\infty,\Omega}$ is the best Lipschitz constant of $f$;

*A differentiable function on an interval is increasing iff $f'\ge0$;

*The Total Differential Theorem in Banach spaces;

*Symmetry of $d^2f(x_0)$, whenever it exists at a point;

*Schwarz-Peano theorem on the inversion of the order of derivations;

*Limit under the sign of derivative;

*Local Inversion and Implicit Function Theorem;

*Every Banach valued continuous curve has an antiderivative; whence, a quick integral theory for continuous Banach valued curves and the possibility of an ODE's theory for Banach valued curves;

*Various estimates for the Taylor expansions.

There are, of course, interesting generalization of the Mean Value Theorem as stated above even in the direction of the existence of the point $x$. Besides Faedo's result you quoted, you can find some more recent papers in the American Mathematical Monthly. But, again, the most fecund ones are in the direction of the inequality "finite increment vs derivative". First, the inequality remains true if $f$ is continuous and differentiable outside  countable set, or also, if it is absolutely continuous and differentiable a.e., whence, respectively, the possibility of a Fundamental Theorem of Calculus for the Cauchy integral (of regulated functions) and for the Lebesgue integral.
