Let $ f \in C^{n+1}([a,b])$ and a polynom p $\in P_{n} $. The support points $ x_{i} = a +ih , i =0,...,n$ are equidistant with $ h \in \mathbb{R} $ so that $ x_{n}=b$.
Show
$ |f(x)-p(x)| \leq h^{n+1}* \frac{||f^{n+1}||_{\infty}}{4(n+1)} \forall x \in [a,b]$.
My first idea is to show this with induction but I don´t get the start. But in lesson we have a corollar that for the interpolation error applies
$ |f(x)-p(x)| \leq |\omega (x)|* max_{\xi \in I}\frac{|f^{n+1}(\xi)|}{(n+1)!} ,x \in I=[a,b]$. With $\omega (x) = \prod_{j=0}^n (x-x_{j})$.
Maybe this could be usefull?
Use the error formula for polynomial interpolation $$ |f(x)-p(x)|\le\frac{|f^{(n+1)}(\xi)|}{(n+1)!}\prod_{k=0}^n|x-x_k| $$ for some $\xi$ inside the interval spanned by the $x_k$ and $x$.
See There is a way to determine error of an interpolation outside of the given range, Error term in polynomial interpolation of non-differentiable function, or similar interpolation error using higher derivatives