Let $f(x,y)=-x^2+2xy+y^3$ , prove that if $p>q \geq 4$, theTaylor expansion of $f$ of order $p$ and $q$ are the same.

Question

For a function $f(x,y)=-x^2+2xy+y^3$ . Prove that if $p>q \geq 4$, then the Taylor expansion of $f$ in any $(x,y) \in \mathbb{R}^{2}$ of order $p$ and $q$ are the same.

For being honest, Im not have done a lot over this problem but the reason is that I really dont know how to attack this problem.

So far, I got that for a function of two variables $f(x,y)$ whose partials all exist to the $n$-th partials at the point $(a,b)$ the Taylor series exapansion is given in the answer of the following question:

Taylor series in two variables

Still, Im worry about developing the Taylor series for the given $p,q$ in the hypothesis and going nowhere just from straight application of the Taylor series expansion definition. I mean, the hypothesis seems strong about $p$ and $q$. Thanks

Answer 1

Hint

What is the amount of $${\partial ^{p+q}f(x,y)\over \partial x^p\partial y^q}$$for $p>q\ge 4$? How do they represent in the Taylor's expansion of $f(x,y)$?