Let $f(x)\in C^{\infty}[a,b]$. For each $x\in [a,b]$ and $n\in\mathbb{N}$, $f^{(n)}(x)\geqslant0$. Prove: $$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n,~\forall x\in[a,b]$$
The hint tells me to evaluate the remainder but I don't know how to do with it exactly. Thanks for your help!
