Math 2.0

statement corollary

A function with zero derivative is constant

Statement

Let II be an open interval and let f ⁣:I→Rf \colon I \to \mathbb{R} be differentiable with f′(x)=0f'(x) = 0 for all x∈Ix \in I. Then ff is constant 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 domain must be an interval. On (0,1)∪(2,3)(0, 1) \cup (2, 3), a function equal to 00 on one part and 11 on the other has zero derivative. So two antiderivatives of a function on an interval differ by a constant. Mathlib states the result as is_const_of_deriv_eq_zero.

Explanations

L1 human

A function whose slope is zero everywhere on an interval never rises or falls. So its graph is a horizontal line.

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To read this node, read these 23 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 zero derivative is constant, statement, no evidence levelA function with zero …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. 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 zero derivative is constant statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Mean_value_theorem, Implications

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