Theorems · Theorem · real analysis
ConvexOn.isMinOn_of_leftDeriv_nonpos_of_rightDeriv_nonneg
∀ {S : Set ℝ} {f : ℝ → ℝ} {x : ℝ},
ConvexOn ℝ S f → x ∈ interior S → derivWithin f (Set.Iio x) x ≤ 0 → 0 ≤ derivWithin f (Set.Ioi x) x → IsMinOn f S x- Defined in
- Mathlib.Analysis.Convex.Deriv
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- Set.ofPredproof · cited by 6,101
- LE.le.transproof · cited by 3,151
- zero_addproof · cited by 2,366
- Set.Ioistatement and proof · cited by 1,463
- Set.Iiostatement and proof · cited by 1,166
- interiorstatement and proof · cited by 714
- not_leproof · cited by 328
- derivWithinstatement and proof · cited by 258
- ConvexOnstatement and proof · cited by 232
- lt_trichotomyproof · cited by 178
Cited by2
Results whose statement or proof uses this declaration.
- ConvexOn.isMinOn_of_rightDeriv_eq_zeroproof · cited by 1
- ConvexOn.isMinOn_of_leftDeriv_eq_zeroproof · cited by 0