Math 2.0

Real functions Convex functions Corollary

A function with nonnegative second derivative is convex

Builds on A differentiable function is convex iff its derivative is increasingA function with nonnegative derivative is increasingSecond derivative

Statement

Let II be an open interval and let f ⁣:I→ℝf \colon I \to \mathbb{R} be twice differentiable. If f′′(x)≥0f''(x) \ge 0 for all x∈Ix \in I, then ff is convex.

Remarks

The derivative f′f' has a nonnegative derivative, so f′f' is increasing. Then convexity follows. If f′′>0f'' > 0 on II, then ff is strictly convex. The converse fails: x4x^4 is strictly convex, but its second derivative at 00 is 00. Mathlib states it as convexOn_of_deriv2_nonneg.

Reading path

To understand this page from the ground up, read these 31 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 functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionFermat'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 …statementDerivative function, definition, no evidence levelDerivative functiondefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementHigher derivative, definition, no evidence levelHigher derivativedefinitionBoundedness theorem, statement, no evidence levelBoundedness theoremstatementSecond derivative, definition, no evidence levelSecond derivativedefinitionExtreme 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…statementA function with nonnegative derivative is increasing, statement, no evidence levelA function with nonne…statementA function with nonnegative second derivative is convex, 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. 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. Differentiable function definition
  17. Fermat's theorem on stationary points statement
  18. Convergent sequence definition
  19. Monotone convergence theorem for sequences statement
  20. Every sequence has a monotone subsequence statement
  21. Derivative function definition
  22. Bolzano–Weierstrass theorem statement
  23. Higher derivative definition
  24. Boundedness theorem statement
  25. Second derivative definition
  26. Extreme value theorem statement
  27. Rolle's theorem statement
  28. Mean value theorem statement
  29. A differentiable function is convex iff its derivative is increasing statement
  30. A function with nonnegative derivative is increasing statement
  31. A function with nonnegative second derivative is convex statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Convex_function, Functions of one variable
About this page
Tag0Z41C17TGP
URIm2://0Z41C17TGP
Kindstatement, corollary
WritingW0 stub
MSC26A51
StatusThis copy of the site has no build data. Run m2 status 0Z41C17TGP.
UsesA differentiable function is convex iff its derivative is increasing, A function with nonnegative derivative is increasing, Second derivative

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