A challenging problem in differential calculus that cannot be solved using conventional methods.

Question

The function f(x) is differentiable on [a, b] and has a third derivative on (a, b).Prove that：there is a $\xi \in(a, b)$,such that $$\begin{array}{l} f(b)=f(a)+\frac{1}{2}(b-a)\left(f^{\prime}(a)+f^{\prime}(b)\right) \\ -\frac{1}{12}(b-a)^{3} f^{\prime \prime \prime}(\xi) \end{array}$$ The question only mentions that it is thrice differentiable on the open interval, and the values of the second-order and higher derivatives at the endpoints may not necessarily exist. Therefore, conventional methods like Taylor expansion or the Cauchy mean value theorem cannot be applied. Hence, this problem is rather challenging. I speculate that I need to apply Darboux's Mean Value Theorem, but I don't know how to use it.