English
There exists c ∈ (a,b) with deriv f c = slope f a b.
Русский
Существует c ∈ (a,b) такое, что производная f в c равна наклону секущей: f′(c) = (f(b)−f(a))/(b−a).
LaTeX
$$$\\exists c\\in I^{o}_{a,b},\\; \\mathrm{deriv}\\ f\\ c = \\dfrac{f(b)-f(a)}{b-a}$$$
Lean4
/-- 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 faster than `C * x` on `D`, i.e., `C * (y - x) < f y - f x` whenever `x, y ∈ D`,
`x < y`. -/
theorem mul_sub_lt_image_sub_of_lt_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
(hf' : DifferentiableOn ℝ f (interior D)) {C} (hf'_gt : ∀ x ∈ interior D, C < deriv f x) :
∀ᵉ (x ∈ D) (y ∈ D), x < y → C * (y - x) < f y - f x :=
by
intro x hx y hy hxy
have hxyD : Icc x y ⊆ D := hD.ordConnected.out hx hy
have hxyD' : Ioo x y ⊆ interior D := subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD⟩
obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD')
have : C < (f y - f x) / (y - x) := ha ▸ hf'_gt _ (hxyD' a_mem)
exact (lt_div_iff₀ (sub_pos.2 hxy)).1 this
