Theorems · Theorem · real analysis

isMinOn_Ici_of_deriv

∀ {f : ℝ → ℝ} {a b : ℝ},
  ContinuousAt f a →
    ContinuousAt f b →
      DifferentiableOn ℝ f (Set.Ioo a b) →
        DifferentiableOn ℝ f (Set.Ioi b) →
          (∀ x ∈ Set.Ioo a b, deriv f x ≤ 0) → (∀ x ∈ Set.Ioi b, 0 ≤ deriv f x) → IsMinOn f (Set.Ici a) b

Suppose f : ℝ → ℝ is continuous at a and b, the derivative f' is nonpositive on Ioo a b and nonnegative on Ioi b. Then f attains its minimum on Ici a at b.

Defined in

Mathlib.Analysis.Calculus.DerivativeTest

Cited by

0 results in Mathlib

Foundations

Depth 192 from the axioms · uses propext, Classical.choice, Quot.sound