Theorems · Theorem · real analysis
isMaxOn_Iic_of_deriv
∀ {f : ℝ → ℝ} {b c : ℝ},
ContinuousAt f b →
ContinuousAt f c →
DifferentiableOn ℝ f (Set.Iio b) →
DifferentiableOn ℝ f (Set.Ioo b c) →
(∀ x ∈ Set.Iio b, 0 ≤ deriv f x) → (∀ x ∈ Set.Ioo b c, deriv f x ≤ 0) → IsMaxOn f (Set.Iic c) bSuppose f : ℝ → ℝ is continuous at b and c, the derivative f' is nonnegative on
Iio b and nonpositive on Ioo b c. Then f attains its maximum on Iic c at b.
- Defined in
- Mathlib.Analysis.Calculus.DerivativeTest
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 192 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Set.Ioostatement and proof · cited by 1,214
- Set.Iiostatement and proof · cited by 1,166
- Set.Iicstatement · cited by 1,111
- ContinuousAtstatement and proof · cited by 697
- derivstatement and proof · cited by 676
- DifferentiableOnstatement and proof · cited by 419
- IsMaxOnstatement · cited by 114
- interior_Iccproof · cited by 37
- convex_Iccproof · cited by 29
- antitoneOn_of_deriv_nonposproof · cited by 25
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