strictConvexOn_pow — Mathlib

English

The function x ↦ √x is strictly concave on [0, ∞).

Русский

Функция x ↦ √x строг concave на [0, ∞).

LaTeX

$$StrictConcaveOn ℝ≥0 univ (NNReal.sqrt)$$

Lean4

/-- `x^n`, `n : ℕ` is strictly convex on the whole real line whenever `n ≠ 0` is even. -/
theorem strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) : StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n :=
  by
  apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
  eta_expand
  simp_rw [deriv_pow_field]
  replace h := Nat.pos_of_ne_zero h
  exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl) (Nat.cast_pos.2 h)

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