Taylor's theorem with Lagrange remainder

Let \(a<b\), \(n\geq0\), and \(c\in[a,b]\). Suppose \(f\colon[a,b]\to\RR\) is \(n\) times continuously differentiable on \([a,b]\) and \(f^{(n+1)}\) exists on \((a,b)\). Let \[ P_n(x)\coloneqq\sum_{k=0}^{n}\frac{f^{(k)}(c)}{k!}(x-c)^k =f(c)+f'(c)(x-c)+\cdots+\frac{f^{(n)}(c)}{n!}(x-c)^n . \] Then for every \(x\in[a,b]\) with \(x\neq c\) there exists \(\xi\) strictly between \(c\) and \(x\) such that \[ f(x)=P_n(x)+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-c)^{n+1}. \] In particular, if \(M\geq0\) and \(\abs{f^{(n+1)}(t)}\leq M\) for all \(t\in(a,b)\), then \[ \abs{f(x)-P_n(x)}\leq\frac{M\abs{x-c}^{n+1}}{(n+1)!}\quad\text{for all } x\in[a,b]. \] [@Rud76].