Math 2.0

Real functions Theorem

A differentiable function is convex iff its derivative is increasing

Builds on Convex functionMean value theoremSlopes of a convex function are increasing

Statement

Let II be an open interval and let f ⁣:I→ℝf \colon I \to \mathbb{R} be differentiable. Then ff is convex if and only if f′f' is increasing on II.

Remarks

If ff is convex, the slope inequalities give f′(x)≤f′(y)f'(x) \le f'(y) for x<yx < y. If f′f' is increasing, the mean value theorem on [x,y][x, y] and [y,z][y, z] gives the slope inequalities. Then the graph lies above each tangent line: f(y)≥f(x)+f′(x)(y−x)f(y) \ge f(x) + f'(x)(y - x). Mathlib states the converse direction as MonotoneOn.convexOn_of_deriv.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Convex function, definition, no evidence levelConvex functiondefinitionLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionSequence, definition, no evidence levelSequencedefinitionSlopes of a convex function are increasing, statement, no evidence levelSlopes of a convex fu…statementLimit 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 differentiable function is convex iff its derivative is increasing, statement, no evidence levelA differentiable func…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. Convex function definition
  2. Limit point of a set definition
  3. Local maximum definition
  4. Sequence definition
  5. Slopes of a convex function are increasing statement
  6. Limit of a function definition
  7. Local minimum definition
  8. Real sequence definition
  9. Subsequence definition
  10. Continuity at a point definition
  11. Derivative definition
  12. Bounded sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Continuous function definition
  16. Fermat's theorem on stationary points statement
  17. Convergent sequence definition
  18. Monotone convergence theorem for sequences statement
  19. Every sequence has a monotone subsequence statement
  20. Bolzano–Weierstrass theorem statement
  21. Boundedness theorem statement
  22. Extreme value theorem statement
  23. Rolle's theorem statement
  24. Mean value theorem statement
  25. A differentiable function is convex iff its derivative is increasing statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Convex_function, Functions of one variable
About this page
Tag005JT76HCK
URIm2://005JT76HCK
Kindstatement, theorem
WritingW0 stub
MSC26A51
StatusThis copy of the site has no build data. Run m2 status 005JT76HCK.
UsesConvex function, Mean value theorem, Slopes of a convex function are increasing

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