IMC 2025 Problem 6

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a continuously differentiable function, and let $b>a>0$ be real numbers such that $f(a)=f(b)=k$ . Prove that there exists a point $\xi \in(a, b)$ such that $$f(\xi)-\xi f^{\prime}(\xi)=k$$