Question 113
Let $n$ be a positive integer. Find all functions $f: [0, \infty) \to \mathbb{R}$, which are $n$ times differentiable, their $n^{th}$ derivative is bounded below and
$$
x \int_{x}^{x+1} f(t) \, dt = \int_{0}^{x} f(t) \, dt,
$$
for every real $x \ge 0$.
$$
x \int_{x}^{x+1} f(t) \, dt = \int_{0}^{x} f(t) \, dt,
$$
for every real $x \ge 0$.