Ideal of germs of functions vanishing at p

Question

Let $M^m$ be a smooth manifold, $p \in M$, and the ideal $I_p=\{f \in C^\infty(M) : f(p)=0 \}$ of the algebra of germs $C^\infty(M)$. Show that:
a) $df_p=0, \forall f \in I^2_p$;
b) $I_p/I^2_p$ is a m-dimensional vector space;
c) $T: T_pM \longrightarrow (I_p/I^2_p)^*$ given by $T(v)(f+I^2_p)=df_p(v)$ defines a linear isomorphism, in which $(I_p/I^2_p)^*$ denotes the dual space of $I_p/I^2_p$.

I proved (a) - too easy - but I'm stuck with (b) and (c). Anyway, if (b) is proven and $T$ is shown to be injective, it's done. But I'm stuck with that too.

Answer 1

Hint: For b) and injectivity in c), use Taylor development around zero in a local chart centered at $p$.