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Cauchy's mean-value theorem in quotient form when the denominator derivative is nonzero
Statement
Let . If are continuous on , differentiable on , and throughout , then and there is such that
Facts & Assumptions
Given: The functions and hypotheses in the statement.
Cauchy's mean value theorem gives for some (Cauchy's mean value theorem: for continuous on with and differentiable on there is with ; no hypothesis on is needed in this product form).
Rolle's theorem says equal endpoint values force an interior zero of the derivative (Rolle's theorem: if , is continuous on , differentiable at every point of , and , then for some ).
Proof
If , Rolle gives with , contrary to the hypothesis. Hence .
Cauchy's theorem supplies with the cross-product identity in [L1].
Divide that identity by the two nonzero factors and to obtain the quotient formula.
Depends on
- Cauchy's mean value theorem: for $f, g$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c \in (a,b)$ with $\bigl(f(b)-f(a)\bigr)g'(c) = \bigl(g(b)-g(a)\bigr)f'(c)$; no hypothesis on $g'$ is needed in this product form
- Rolle's theorem: if $a < b$, $f$ is continuous on $[a,b]$, differentiable at every point of $(a,b)$, and $f(a) = f(b)$, then $f'(c) = 0$ for some $c \in (a,b)$
- The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set
- The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$
Used by
Dependency tree · two levels
24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis I, Taylor's theorem and related calculus (standard reference, not scraped)
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)
- J. Lebl, Basic Analysis I, Mean value theorem (standard reference, not scraped)
- Colgate University MATH 323, Chapter 5 notes (standard reference, not scraped)