Theorems · Theorem · real analysis

StrictConvexOn.slope_lt_of_hasDerivWithinAt_Iio

∀ {S : Set ℝ} {f : ℝ → ℝ} {x y f' : ℝ},
  StrictConvexOn ℝ S f → x ∈ S → y ∈ S → x < y → HasDerivWithinAt f f' (Set.Iio y) y → slope f x y < f'

If f : ℝ → ℝ is strictly convex on S and differentiable at y ∈ S, then the slope of any secant line with right endpoint at y is strictly less than the left derivative at y.

Defined in

Mathlib.Analysis.Convex.Deriv

Cited by

4 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound