Rolle's theorem
Rolle's theorem guarantees that a differentiable curve with equal values at the ends of an interval has at least one interior point with a horizontal tangent, provided it is continuous on the closed interval and differentiable on its interior. It is a special case of the mean value theorem and a foundational result in differential calculus.
Statement
Let f be a real-valued function that is continuous on a closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there is at least one c in (a, b) such that f′(c) = 0. The conclusion says that the derivative vanishes somewhere strictly between the endpoints; it does not say that there is exactly one such point.
Geometric interpretation
If a smooth graph starts and ends at the same height, then somewhere between those endpoints it has a horizontal tangent. If the function is constant on the interval, its derivative is zero throughout the interior. Otherwise, continuity ensures that a maximum or minimum is attained; when an interior extremum occurs at a point where the function is differentiable, Fermat's theorem says its derivative there is zero.
Why the conditions matter
Continuity on the whole closed interval and differentiability at every interior point are essential hypotheses. For example, the absolute-value function, defined by f(x) = −x for x < 0 and f(x) = x for x ≥ 0, is continuous on [−1, 1] and has f(−1) = f(1), but it is not differentiable at its minimum x = 0. Its derivative is −1 for x < 0 and 1 for x > 0, so no interior point satisfies f′(x) = 0. This illustrates why equal endpoint values alone do not guarantee the conclusion.
Relation to the mean value theorem
The mean value theorem states that for a function continuous on [a, b] and differentiable on (a, b), some interior point c satisfies f′(c) = (f(b) − f(a))/(b − a). When f(a) = f(b), the secant slope on the right is zero, giving Rolle's theorem. Conversely, Rolle's theorem is used in standard proofs of the mean value theorem by subtracting the line through the graph's endpoints from the function and applying Rolle's result to the difference.
Uses
Rolle's theorem helps establish further results in calculus and real analysis. For instance, if a differentiable function has two distinct zeros, applying Rolle's theorem between them shows that its derivative has a zero between those zeros. Repeated applications can relate the number and locations of zeros of a function to those of its derivatives, subject to the required continuity and differentiability conditions.
Key facts
- Hypotheses
- Continuity on [a, b], differentiability on (a, b), and equal endpoint values f(a) = f(b).
- Conclusion
- At least one interior point c satisfies f′(c) = 0.
- Mean value theorem
- Rolle's theorem is the equal-endpoint special case of the mean value theorem.
- Critical condition
- Differentiability at interior points cannot be omitted; a cusp can violate the conclusion.
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References
- 1.4.4 The Mean Value Theorem
OpenStax, Rice University · other · reliability: unverified · retrieved 30 October 2026
Supports: States and proves Rolle's theorem, explains the need for differentiability, relates it to the mean value theorem, and gives calculus examples.
- 2.4.4: Rolle’s Theorem and The Mean Value Theorem
Mathematics LibreTexts, Mission College Calculus I · other · reliability: unverified · retrieved 30 October 2026
Supports: Gives the precise hypotheses and conclusion, a proof via extrema, an example where a nondifferentiable function fails the conclusion, and the connection to the mean value theorem.
- 3.Calculus I – The Mean Value Theorem
Paul's Online Math Notes, Lamar University · other · reliability: unverified · retrieved 30 October 2026
Supports: Sets out Rolle's theorem, explains how it is used in the proof of the mean value theorem, and demonstrates an application involving polynomial roots.
- 4.Math 221 – Notes on Rolle’s Theorem, The Mean Value Theorem, l’Hôpital’s rule, and the Taylor-Maclaurin formula
University of Wisconsin–Madison, Mathematics Department · other · reliability: unverified · retrieved 30 October 2026
Supports: University course notes state Rolle's theorem and illustrate its use in results about derivatives, l’Hôpital's rule, and Taylor's formula.
References are suggested by the research engine and may be incomplete. Unverified URLs are omitted rather than guessed.
Article history
- Version 230 October 2026 · EditorUpdated by Zapia
- Version 130 October 2026 · EditorCreated by Zapia