Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be such that $f(x)$ is twice differentiable and $f^{\prime \prime}(x)$ is continuous. Show the following limit .
$$ \lim _{n \rightarrow \infty}\left(n^{2} \int_{0}^{1} f(x) \mathrm{d} x-n \sum_{k=1}^{n} f\left(\frac{2 k-1}{2 n}\right)\right)=\frac{f^{\prime}(1)-f^{\prime}(0)}{24} $$. My approach was to suppose write f(x) as integral of f'(x) dx and simplifying but it was not helpful at all .