Who was the first to prove the mean value theorem?

Question

Who was the first to prove the mean value theorem, i.e., the existence of an intermediate point where the slope equals the average slope over the interval?

Answer 1

As per your comment requesting details of the linked paper :

Besenyei's paper begins with a history of the development of Rolle's Theorem into its general form. It follows this with the history of various generalisations of Rolle's Theorem equivalent to MVT and attributed to the likes of Cauchy, Bonnet, Serret, Dini, and Harnack.

Looking at Besenyei's paper, at first it would appear that Joseph Alfred Serret is the first to state and prove the result in its modern form, however this is not the case - please see below. From Serret's Cours de calcu infinitesimal of 1868 as Theorem I (translation) :

Let $f(x)$ be a function of $x$ which remains continuous for values of $x$ between two given limits, and which, for these values, has a well-determined derivative $f'(x)$. If $x_0$ and $X$ denote two values of $x$ between these same limits, the following $$\frac{f(X) - f(x_0)}{X-x_0} = f'(x_1)$$ will hold, with $x_1$ being a value between $x_0$ and $X$.

Besenyei's paper does not include the relevant proof.

This appears on the page denoted 11/15 (lower right hand corner) - page 83/152 of the PDF document.

However, according to this paper by Renaud Chorlay, Serret is here simply stating Bonnet's proof. (See page 7 of 16 of the PDF document.) Thus, the correct answer as to who first proved MVT in its modern form appears to be Pierre-Ossian Bonnet. The Renaud Chorlay paper does appear to give Bonnet's proof.

This conclusion appears to be contrary to the wikipedia entry for the MVT which credits Cauchy as the first to state it in its modern form. However, Cauchy's statement of MVT is actually and extension of MVT to deal with two different continuous functions $f$ and $g$ which states $$\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}$$ and which predates the Serret/Bonnet version of MVT.

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EDIT

As Mikhail has pointed out in his comment, the correct answer is in fact Cauchy, as is evident in Besenyei's paper. Cauchy gives both forms of stating the MVT.

Answer 2

The theorem is black on white in Lagrange’s Théorie des fonctions analytiques, as the initial case of “Taylor’s theorem with Lagrange remainder” (1797, §§51–53):

Hence the value of the quantity $P$ relative to $z=1$ [i.e., $\frac{f(x)\ -\ f(0)}x$] can be expressed as $f'u$, $u$ being a quantity between $0$ and $x$. Likewise (...)

52. Whence there results finally this new theorem, remarkable in its simplicity and its generality, that denoting by $u$ a quantity unknown but comprised between the limits $0$ and $x$, one can successively develop any function of $x$ and of arbitrary other quantities according to the powers of $x$, thus (...) \begin{align} f(z+x) &= fz + x\,f'(z+u),\\ &=fz + x\,f'z + \frac{x^2}2\,f''(z+u),\\ &=fz + x\,f'z + \frac{x^2}2\,f''z + \frac{x^3}{2.3}\,f'''(z+u),\\ &\ \ \&\mathrm c. \end{align}

Lagrange also spells out the “Rolle” case in his next book (1798, pp. 166, 175):

Consideration of the maxima and minima of parabolic lines led Stirling to a method for counting and bounding real roots in degrees 3 and 4, which Euler generalized in his Differential calculus [§298]. This method boils down, basically, to Rolle’s (... :)

between two consecutive real roots of the equation $\mathrm F\,x=0$, there always falls a real root of the equation $\mathrm F'x=0$.

So I question the accepted attribution to Cauchy (Wikipedia 2007, 2016) — though of course it all hinges on: “first to prove” under what assumptions, and how rigorously? For discussion of that: e.g.

• Dugac, Pierre, Histoire du théorème des accroissements finis, Arch. Int. Hist. Sci. 30, 86-101 (1980). ZBL0447.26003.

• Grabiner, Judith, Review of “Histoire du théorème des accroissements finis” by Pierre Dugac. Paris (Université Pierre et Marie Curie). 1979. 60 pp. Historia Math. 8, 214-219 (1981).

• Persson, Lars-Erik; Rafeiro, Humberto; Wall, Peter, Historical synopsis of the Taylor remainder, Note Mat. 37, 1-21 (2017). MR3733800.

Edit as requested:

Briefly, Dugac contends (and repeats in a later paper and book) that 1º) Lagrange first singled out, stated, and proved — perhaps inconclusively — the mean-value property for analytic functions; 2º) he was to an extent foreshadowed by Cavalieri (1635, p. 19); 3º) the first conclusive or “rigorous” proof in print is by Dini (1878, p. 71). To me this is all rather noncontroversial: the book I learned it from called it “Lagrange’s theorem”. So do quite a few others. Current textbooks (Thomas, Stewart, Petrovic,...) credit it to Lagrange. So do Bottazzini (1986, p. 53), Grabiner (1981, p. 122), Edwards (1979, p. 313), Cajori (1910, pp. 308-311), the Encyklopädie (1899, II A 2 §§7, 11), and so on all the way to Lacroix (1810: vol. 1, pp. liv, 385-386; 1800: vol. 3, pp. vii, 384). I am beginning to wonder if Besenyei might be the only author who doesn’t attribute the theorem to Lagrange?

That said, it is true that a wide shift of emphasis happened between the way Lagrange thought of his theorem and the way we use (and prove) it now. Besides already containing most of the above story, the French Encyclopédie article (1912, II 3 §§11, 22) says this well in footnote 116:

From Rolle’s theorem¹¹⁴⁾ one readily deduces¹¹⁵⁾ the mean value formula which plays a basic role throughout Calculus¹¹⁶⁾ and is also, like Rolle’s theorem, basically a mere translation into precise language of the intuitive fact that B. Cavalieri had brought to the geometers’ attention.

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^{114) On the first rigorous proofs of Rolle’s theorem, see A. Pringsheim (1900, p. 454); see also U. Dini (1878, p. 75); A. Harnack (1881); M. Pasch (1882); P. Mansion (1887); J. Tannery (1886); A. Demoulin (1902).}

^{115) In his lectures at École polytechnique, O. Bonnet fully rigorously demonstrated this formula on the basis of Rolle’s theorem. He also gave a proof of Rolle’s theorem which, at least as exposed in J. A. Serret (1868, pp. 17-19), is far from immune to objections. (...)}

^{116) One should note that this fundamental role stems mainly from the mode of exposition of Calculus generally adopted today. For J. L. Lagrange, who took a different point of view, this formula was merely a consequence of Taylor’s formula (1797, p. 49).}

To summarize, my take for now (based on these references and subject to your corrections):

1635:  Cavalieri states MVT hence Rolle. (Proof rejected by Guldin, 1640.)
1690:  Rolle proves Rolle for polynomials.
1755:  Euler states Rolle for differentiable functions. (Sketch of proof accepted at the time.)
1797:  Lagrange proves MVT (hence Rolle) for analytic functions. (Proof accepted at the time.)
1823:  Cauchy proves MVT (hence Rolle) for C¹ functions. (Proof essentially same as Lagrange.)
1868:  Bonnet rearranges proof of MVT to rely on Rolle as a lemma.
1878:  Dini proves Rolle (hence MVT now) for differentiable functions. (Our current proof.)