Real function/Taylor formula/(n+1)-times continuously differentiable/Lagrange/Fact

Taylor formula

Let ${}I$ denote a real interval,

$f\colon I\longrightarrow \mathbb {R}$

an ${}(n+1)$-times differentiable function, and ${}a\in I$ an inner point of the interval.

Then for every point

${}x\in I$, there exists some ${}c\in I$ such that

$f(x)=\sum _{k=0}^{n}{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}+{\frac {f^{(n+1)}(c)}{(n+1)!}}(x-a)^{n+1}.$

Here, ${}c$ may be chosen between ${}a$ and ${}x$.

Write a proof