Math 2.0

statement proposition

First derivative test

Statement

Let ff be a real function on an open interval (c−δ,c+δ)(c - \delta, c + \delta) with δ>0\delta > 0, continuous at cc and differentiable at each x≠cx \ne c. If f′(x)≤0f'(x) \le 0 for c−δ<x<cc - \delta < x < c and f′(x)≥0f'(x) \ge 0 for c<x<c+δc < x < c + \delta, then ff has a local minimum at cc. If the inequalities are reversed, then ff has a local maximum at cc.

Remarks

The function is decreasing on the left and increasing on the right of cc. The test needs no derivative at cc. It works for ∣x∣|x| at 00. Mathlib states the minimum case as isLocalMin_of_deriv.

Explanations

L1 human

If the graph goes down until cc and then goes up, the point cc is the bottom of a valley.

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What this needs

To read this node, read these 24 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 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…statementFirst derivative test, statement, no evidence levelFirst 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. 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
  24. First derivative test statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Derivative_test, Precise statement of first-derivative test

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