Math 2.0

Real functions Differentiation Theorem

Rolle's theorem

Builds on Extreme value theoremFermat's theorem on stationary points

Statement

Let a<ba < b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be continuous on [a,b][a, b] and differentiable on (a,b)(a, b), with f(a)=f(b)f(a) = f(b). Then there is c∈(a,b)c \in (a, b) with f′(c)=0f'(c) = 0.

Remarks

By the extreme value theorem, ff has a maximum and a minimum on [a,b][a, b]. If both occur at the endpoints, ff is constant. Otherwise one occurs at an interior point cc, and Fermat's theorem gives f′(c)=0f'(c) = 0. Differentiability inside the interval is needed. The function ∣x∣|x| on [−1,1][-1, 1] has no point with derivative 00. Mathlib states it as exists_deriv_eq_zero.

History

Michel Rolle proved a version for polynomials in 1691. Cauchy gave a proof of the general theorem in 1823.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementRolle's theorem, statement, no evidence levelRolle's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. Bounded sequence definition
  11. Limit of a sequence definition
  12. Monotone sequence definition
  13. Continuous function definition
  14. Fermat's theorem on stationary points statement
  15. Convergent sequence definition
  16. Monotone convergence theorem for sequences statement
  17. Every sequence has a monotone subsequence statement
  18. Bolzano–Weierstrass theorem statement
  19. Boundedness theorem statement
  20. Extreme value theorem statement
  21. Rolle's theorem statement

Where this leads

112 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Rolle%27s_theorem, Statement
About this page
Tag0194DY35BG
URIm2://0194DY35BG
Kindstatement, theorem
WritingW1 stated
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0194DY35BG.
UsesExtreme value theorem, Fermat's theorem on stationary points
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