Theorems · Theorem · real analysis

Convex.mul_sub_le_image_sub_of_le_deriv

∀ {D : Set ℝ},
  Convex ℝ D →
    ∀ {f : ℝ → ℝ},
      ContinuousOn f D →
        DifferentiableOn ℝ f (interior D) →
          ∀ {C : ℝ}, (∀ x ∈ interior D, C ≤ deriv f x) → ∀ x ∈ D, ∀ y ∈ D, x ≤ y → C * (y - x) ≤ f y - f x

Let f be a function continuous on a convex (or, equivalently, connected) subset D of the real line. If f is differentiable on the interior of D and C ≤ f', then f grows at least as fast as C * x on D, i.e., C * (y - x) ≤ f y - f x whenever x, y ∈ D, x ≤ y.

Defined in

Mathlib.Analysis.Calculus.Deriv.MeanValue

Cited by

3 results in Mathlib

Foundations

Depth 189 from the axioms · uses propext, Classical.choice, Quot.sound