lean_workbook_plus_38953

Consider a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(2x + 1)^2 - 1 = 2f(2x)f(x + 1)$. Prove that there exists some $x \in \mathbb{R}$ such that $f(2x + 1) \ge f(x)$.