Math 2.0

statement theorem

Rolle's theorem

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.

Explanations

L1 human

A smooth graph that starts and ends at the same height must turn around somewhere. At the turning point, the tangent is horizontal.

What this node needs for W2
  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

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To read this node, read these 21 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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

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75 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Rolle%27s_theorem, Statement

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