Theorems · Theorem · real analysis

concaveOn_iff_slope_anti_adjacent

∀ {𝕜 : Type u_1} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜},
  ConcaveOn 𝕜 s f ↔
    Convex 𝕜 s ∧ ∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)

A function f : 𝕜 → 𝕜 is concave iff for any three points x < y < z the slope of the secant line of f on [x, y] is greater than the slope of the secant line of f on [y, z].

Defined in

Mathlib.Analysis.Convex.Slope

Cited by

0 results in Mathlib

Foundations

Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRing