Math 2.0

Real functions Corollary

A function with nonnegative derivative is increasing

Builds on Mean value theorem

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) \ge 0 for each interior point xx of II, then ff is increasing on II. That is, f(x)≤f(y)f(x) \le f(y) for x≤yx \le y in II.

Remarks

The proof uses the mean value theorem. For differentiable ff on an open interval, the converse holds too: if ff is increasing, then f′≥0f' \ge 0. Mathlib states it as monotoneOn_of_deriv_nonneg.

Reading path

To understand this page from the ground up, read these 23 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 theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementA function with nonnegative derivative is increasing, statement, no evidence levelA function with nonne…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. 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
  23. A function with nonnegative derivative is increasing statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Monotonic_function, Some basic applications and results
About this page
Tag026BJBQNGJ
URIm2://026BJBQNGJ
Kindstatement, corollary
WritingW0 stub
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 026BJBQNGJ.
UsesMean value theorem

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