Math 2.0

statement corollary

A function with positive derivative is strictly increasing

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.

Explanations

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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 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. 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

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Sources

  1. Web https://en.wikipedia.org/wiki/Monotonic_function, Some basic applications and results

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