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The Generalized Mean-Value Theorem or Cauchy's theorem or of the finite increments theorem says:

If functions $f$ and $g$ are both continuous on $[a,b]\subset\Bbb R$ and differentiable on $]a,b[\subset\Bbb R$, and if $g'(x)\neq 0$ for every $x\in]a,b[$, then there exists a number $\xi$ in $]a,b[$ such that

$$\frac{f(b)-f(a)}{g(a)-g(b)}=\frac{f'(\xi)}{g'(\xi)}$$

If $g(x)=x$ I found the Lagrange's theorem.

https://en.m.wikipedia.org/wiki/Mean_value_theorem#Proof_of_Cauchy.27s_mean_value_theorem

This theorem says that it is also possible that there are multiple tangents parallel to the secant (see the image).

  • What is the importance to have multiple tangents parallel to the secant in a specific exercise?

This theorem is now almost omitted from Italian school books: there is only the enunciate and few exercises.

I have taken an screenshot of a English book,

enter image description here

and I see many exercises of Lagrange's theorem to justify inequalities.

  • At this point I ask to all users on Math.SE a curiosity: what could be good uses for the generalized mean-value theorem or of Cauchy?
Arctic Char
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Sebastiano
  • 7,649
  • You can use that result to justify L'Hostpital rule. You can also use the result to get Taylor's polynomial (with some additional assumptions) – Mittens Aug 10 '20 at 22:58
  • @OliverDiaz Yes of course. There is into the link of the 1st image to justify de l'Hopital rule. Every answer is welcome for the Taylor's polynomial. – Sebastiano Aug 10 '20 at 23:00
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    For Taylor's polynomial here is a link that uses the GMVT – Mittens Aug 10 '20 at 23:04

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