Math 2.0

statement proposition

Second derivative test

Statement

Let ff be a real function differentiable on an open interval around cc, with f′(c)=0f'(c) = 0. Assume that the second derivative f′′(c)f''(c) exists. If f′′(c)>0f''(c) > 0, then ff has a strict local minimum at cc. If f′′(c)<0f''(c) < 0, then ff has a strict local maximum at cc.

Remarks

If f′′(c)>0f''(c) > 0, then f′(x)/(x−c)→f′′(c)f'(x)/(x - c) \to f''(c). So f′f' is negative just left of cc and positive just right of cc, and the first derivative test applies. If f′′(c)=0f''(c) = 0, the test gives no answer. The functions x4x^4, −x4-x^4, and x3x^3 at 00 show all three cases. Mathlib states the minimum case as isLocalMin_of_deriv_deriv_pos.

Explanations

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What this needs

To read this node, read these 29 nodes in this order. The foundations come first, and the list ends with this node.

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 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 function with nonnegative derivative is increasing, statement, no evidence levelA function with nonne…statementFirst derivative test, statement, no evidence levelFirst derivative teststatementSecond derivative test, statement, no evidence levelSecond derivative teststatement
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.
  • 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. Differentiable 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. Derivative function definition
  20. Bolzano–Weierstrass theorem statement
  21. Higher derivative definition
  22. Boundedness theorem statement
  23. Second derivative definition
  24. Extreme value theorem statement
  25. Rolle's theorem statement
  26. Mean value theorem statement
  27. A function with nonnegative derivative is increasing statement
  28. First derivative test statement
  29. Second derivative test statement

Sources

  1. Web https://en.wikipedia.org/wiki/Derivative_test, Second-derivative test (single variable)

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