Let $f\colon [a,b]\to\Bbb R$ be a continuous function on $[a,b]$ and $f''(x)$ exists for all $x\in (a,b)$. Let $a<c<b$, then there exists a point $\xi$ in $(a,b)$ such that $$ f(c)=\frac{b-c}{b-a}f(a)+\frac{c-a}{b-a}f(b)+\frac12(c-a)(c-b)f''(\xi).$$
In this problem, instantaneously I am thinking of applying Rolle s ' theorem.. Am I correct in my thought approach? If not kindly provide a way out how to solve this problem elegantly..thank you . Even in Rolle s how can I go about.... The RHS seems so frightening to prove ......
Any ideas how to go about this problem ...anyone