theorem extracted_formal_statement_4 (x τ : ℝ) (hx : |x| ≤ 1) (hτ : 0 ≤ τ) (hx_le_1 : x ≤ 1) :
  let f := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ));
  f 0 = 0 → f τ ≤ 0 → f τ ≤ f 0 := sorry

State before:
case a
x τ : ℝ
hx : |x| ≤ 1
hτ : 0 ≤ τ
hx_le_1 : x ≤ 1
f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
hf0 : f 0 = 0
hf'0 : f' 0 = 0
⊢ f τ ≤ 0

Tactic: rw [← hf0]

State after:
[case a
 x τ : ℝ
 hx : |x| ≤ 1
 hτ : 0 ≤ τ
 hx_le_1 : x ≤ 1
 f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
 f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
 f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
 hf0 : f 0 = 0
 hf'0 : f' 0 = 0
 ⊢ f τ ≤ f 0]

Relevant lemmas: