deriv_deriv_nonneg_of_continuousAt

Plain-English statement

- At a local minimum of a continuous real function, the second derivative, when represented by `deriv (deriv f)`, cannot be negative. This complements Mathlib's sufficient strict second-derivative test with the necessary non-strict direction.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem IsLocalMin.deriv_deriv_nonneg_of_continuousAt
    {f : ℝ → ℝ} {x : ℝ} (hmin : IsLocalMin f x)
    (hc : ContinuousAt f x) :
    0 ≤ deriv (deriv f) x := by
  by_contra hnonneg
  have hneg : deriv (deriv f) x < 0 := lt_of_not_ge hnonneg
  have hmax : IsLocalMax f x :=
    isLocalMax_of_deriv_deriv_neg hneg hmin.deriv_eq_zero hc
  have hconst : f =ᶠ[𝓝 x] fun _ : ℝ => f x := by
    filter_upwards [hmin, hmax] with y hymin hymax
    exact le_antisymm hymax hymin
  have hderiv : deriv f =ᶠ[𝓝 x] fun _ : ℝ => 0 := by
    simpa using hconst.deriv
  have hzero : deriv (deriv f) x = 0 := by
    calc
      deriv (deriv f) x = deriv (fun _ : ℝ => 0) x := hderiv.deriv_eq
      _ = 0 := by simp
  linarith

/-- At a local maximum of a continuous real function, the second derivative,
when represented by `deriv (deriv f)`, is nonpositive. -/

Open AutoSamplingTheory/TechnicalLemmas/Geometry/SphereSecondOrder.lean:24published source at ed709a7f320d

Proof architecture

Reusable second-order sign condition for optimization and maximum-principle arguments.

Lean proof walkthrough

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`exact` closes the current goal with an already typed term.
`simpa` closes the goal after a controlled simplification of a typed result.
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Hidden assumptions and non-claims

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