Math 2.0

Real functions Corollary

A function with positive derivative is strictly increasing

Builds on Mean value theoremStrictly monotone function

Statement

Let II be an interval and let f ⁣:I→ℝf \colon I \to \mathbb{R} be continuous, and differentiable at each interior point of II. If f′(x)>0f'(x) > 0 for each interior point xx of II, then ff is strictly increasing on II.

Remarks

For x<yx < y in II, the mean value theorem gives f(y)−f(x)=f′(c)(y−x)>0f(y) - f(x) = f'(c)(y - x) > 0. The converse fails: x3x^3 is strictly increasing, but its derivative at 00 is 00. Mathlib states it as strictMonoOn_of_deriv_pos.

Reading path

To understand this page from the ground up, read these 24 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 levelSequencedefinitionStrictly monotone function, definition, no evidence levelStrictly monotone fun…definitionLimit 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 theoremstatementA function with positive derivative is strictly increasing, statement, no evidence levelA function with posit…statement
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. Strictly monotone function definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Real sequence definition
  8. Subsequence definition
  9. Continuity at a point definition
  10. Derivative definition
  11. Bounded sequence definition
  12. Limit of a sequence definition
  13. Monotone sequence definition
  14. Continuous function definition
  15. Fermat's theorem on stationary points statement
  16. Convergent sequence definition
  17. Monotone convergence theorem for sequences statement
  18. Every sequence has a monotone subsequence statement
  19. Bolzano–Weierstrass theorem statement
  20. Boundedness theorem statement
  21. Extreme value theorem statement
  22. Rolle's theorem statement
  23. Mean value theorem statement
  24. A function with positive derivative is strictly increasing statement

Where this leads

30 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Monotonic_function, Some basic applications and results
About this page
Tag07J5X2T1ZX
URIm2://07J5X2T1ZX
Kindstatement, corollary
WritingW0 stub
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 07J5X2T1ZX.
UsesMean value theorem, Strictly monotone function

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