Math 2.0

Real functions Differentiation Theorem

Mean value theorem

Builds on 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). Then there is c∈(a,b)c \in (a, b) with f′(c)=f(b)−f(a)b−a.f'(c) = \frac{f(b) - f(a)}{b - a}.

Remarks

Apply Rolle's theorem to g(x)=f(x)−f(b)−f(a)b−a(x−a)g(x) = f(x) - \frac{f(b) - f(a)}{b - a} (x - a), which has g(a)=g(b)g(a) = g(b). It turns bounds on f′f' into bounds on ff. It gives zero derivative implies constant and the monotonicity tests. Mathlib states it as exists_deriv_eq_slope.

History

Augustin-Louis Cauchy stated and proved the theorem in its modern form in 1823.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
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 theoremstatementMean value theorem, statement, no evidence levelMean value 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.
  • no evidence level
  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
  22. Mean value theorem statement

Where this leads

103 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Mean_value_theorem, Statement
About this page
Tag0BARQY52N5
URIm2://0BARQY52N5
Aliaseslagrange-mean-value-theorem
Kindstatement, theorem
WritingW1 stated
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0BARQY52N5.
UsesRolle's theorem
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