Abstract
Theorem 1.1 (Cauchy’s Mean-Value Theorem). If f and g are real-valued functions of a real variable, both continuous on the bounded closed interval [a,b], differentiable in the extended sense on (a; b) with g′(x) ≠ 0 for x ∈ (a; b), having derivatives which are not simultaneously infinite, then (1) g(a) ≠ g(b); (2) there exists an x0 ∈ (a; b) such that
(3) if f(a) ≠ f(b), then at the x0 in (1.1), f′(x0) and g′(x0) are both finite.
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© 1983 Springer-Verlag New York, Inc.
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Fischer, E. (1983). L’Hôpital’s Rule—Taylor’s Theorem. In: Intermediate Real Analysis. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9481-5_9
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DOI: https://doi.org/10.1007/978-1-4613-9481-5_9
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