I recently ran into the following exercise:
If $R(x)$ is a $C^\infty$-function near the origin in $\mathbb{R}^n$, satisfying $R(0)=0$ and $DR(0)=0$, show that there exist smooth functions $r_{jk}(x)$ such that$R(x)=\sum r_{jk}(x)x_jx_k.$
I know that the fundamental theorem of calculus applied to $\varphi(t)=F(x+ty)$ yields$F(x+y)=F(x)+\int_0^1DF(x+ty)y\text{ }dt,$provided that $F$ is $C^1$. Using this result, we can write $R(x)=\Phi(x)x$, $\Phi(x)=\int_0^1DR(tx)\text{ }dt$, since $R(0)=0$. Then $\Phi(0)=DR(0)=0$, so we can apply it again to obtain $\Phi(x)=\Psi(x)x$.
I get stuck here, however. Do you guys have any ideas on how to begin to prove that such smooth functions exist? Thanks in advance.