Ideal of smooth function on a manifold vanishing at a point I'm trying to prove the following lemma: let $M$ be a smooth manifold and consider the algebra $C^{\infty}(M)$ of smooth functions $f\colon M \to \mathbb{R}$. Given $x_0 \in M$, consider the ideals
$$\mathfrak{m}_{x_0} := \{f\in C^{\infty}(M) : f(x_0)=0\},$$
$$\mathfrak{I}_{x_0} := \{f\in C^{\infty}(M) : f(x_0)=0, df(x_0)=0\}.$$
Then $\mathfrak{I}_{x_0} = \mathfrak{m}^2_{x_0}$, i.e. any function $f$ vanishing at $x_0$ together with its derivatives can be written as
$$f=\sum_kg_kh_k, \quad g_k, h_k \in \mathfrak{m}_{x_0}.$$
I have no idea about how to prove the inclusion $\mathfrak{I}_{x_0} \subseteq \mathfrak{m}^2_{x_0}$.

$\forall i \in \{1,2,\dots,n\}$, let $g_i(x_1, \dots, x_n)=\int_0^1\frac{\partial f}{\partial x_i}(tx_1, \dots, tx_n)dt$, it is easy to verify that $f=\sum_{i=1}^nx_ig_i$
