Let $f: U \to \mathbb{R}$ of class $C^{k}$ ($i \leq k \leq \infty$) in the convex open $U \subset \mathbb{R}^{2}$ containing $(0,0)$. Suppose that $f$ all partial derivatives of order $\leq i$ vanish in $(0,0)$. Prove that there exists functions $a_{0},..., a_{i}: U \to \mathbb{R}$ of class $C^{k-1}$, such that $$f(x,y) = \sum_{j=0}^{i}a_{j}(x,y)x^{j}y^{i-j}. \quad \forall (x,y) \in U$$
I didn't have a good idea to start solving, but I don't want a complete solution, I just want some hints to get started. My professor hint me to use Taylor Formula, I tried to do this, but I couldn't see how to use.
Consider function $g(t) = f(tx,ty)$ with parameter $t\in[0,1]$. We have (since $g(0) = f(0,0) = 0$) $$ f(x,y) = g(1)-g(0) = \int_0^1g'(t)dt = \int_0^1\left(x\partial_x+y\partial_y\right)f(tx,ty)dt= xc_0(x,y) + yc_1(x,y), $$ where functions $$ c_0(x,y) = \int_0^1\partial_xf(tx,ty)dt, \quad c_1(x,y) = \int_0^1\partial_yf(tx,ty)dt. $$ are of class $C^{k-1}$. Note that we've just proved so-called Hadamard's lemma.
Since all the first partial derivatives of $f$ vanish in zero, $c_0(0,0) = c_1(0,0) = 0$, and we may apply above result to them: $$ c_0(x,y) = xc_{00}(x,y) + yc_{01}(x,y), \quad c_1(x,y) = xc_{10}(x,y) + yc_{11}(x,y) $$ where $c_{ij}$ are function of class $C^{k-2}$. Repeating this procedure $i-1$ times and collecting all the results we'll obtain reminded expansion $$ f(x,y) = \sum_{j=0}^{i}C_j(x,y)x^jy^{i-j}, $$ where $C_j(x,y) = \sum_{|\alpha|=j} c_\alpha(x,y)$ are of class $C^{k-i}$ ($\alpha$ is multi-index of dim $i$).