A Proof of Bonnet's Version of the Mean Value Theorem by Methods of Cauchy

Abstract

The proof of the mean value theorem for differentiable functions presented in modern calculus texts is due to Bonnet (1860s) and depends in an essential way on the extreme value property for continuous functions proved by Weierstrass (1861). This proof of the mean value theorem has intuitive quality but mainstream texts omit the proof of Weierstrass's result on account of its difficulty, thereby depriving the reader of a complete proof of the mean value theorem. Forty years earlier, Cauchy proved the intermediate value property for continuous functions and made a significant but flawed effort to prove a mean value inequality. This note presents a proof of the modern mean value theorem using methods of Cauchy. The key geometrical insight is provided by a simple inequality involving the slopes of sides of a triangle whose vertices lie in the graph of the function. This inequality motivates the use of the intermediate value property in the proof of the mean value theorem and facilitates final determination of the required value of the derivative. An advantage of this alternative to Bonnet's proof is that it is based directly on the completeness of the real numbers rather than on the result of Weierstrass.