Theorems · Theorem · real analysis

ConvexOn.slope_mono

∀ {𝕜 : Type u_1} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} {x : 𝕜},
  ConvexOn 𝕜 s f → x ∈ s → MonotoneOn (slope f x) (s \ {x})

If f : 𝕜 → 𝕜 is convex on s, then for any point x ∈ s the slope of the secant line of f through x is monotone on s \ {x}.

Defined in

Mathlib.Analysis.Convex.Deriv

Cited by

8 results in Mathlib

Foundations

Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRing