Question 10
Let $f : [0, 1] \to \mathbb{R}$ be an integrable function. If $f(1) = 0$ and $f$ is differentiable in $1$, prove that
$$
\lim_{n \to \infty} n^2 \int_0^1 x^n f(x) dx = -f'(1).
$$
$$
\lim_{n \to \infty} n^2 \int_0^1 x^n f(x) dx = -f'(1).
$$