Need some help in this question!
Let $f(x,y)$ be $C^3$ in the open set $A\subset \Bbb{R}^2$ and let $(x_0,y_0)$ be a point of $A$. Proof that there are an open ball $B$ of center $(x_0,y_0)$, with $B\subset A$, and a number $M>0$ such that, for all $(x,y)\in B$
$$|f(x,y)-P_2(x,y)|\leq M \|(x,y)-(x_0,y_0)\|^3$$
where $P_2(x,y)$ is the Taylor polynomial of order 2 of $f$ around $(x_0,y_0)$.
This appeared to me like an exercise. Is this a famous result? Has some special name? I'm grateful for any help!
This is a consequence of Taylor's theorem - something like the Wikipedia article should be helpful.