Maximal ideals of rings of continuous functions on compact Hausdorff spaces

For a topological space \(X\), let \(C(X)\) be the ring of continuous real-valued functions on \(X\). For \(f\in C(X)\) and \(S\subseteq X\), define \[ Z(f)=\{p\in X:f(p)=0\}, \qquad I(S)=\{f\in C(X):S\subseteq Z(f)\}. \]

1. Show that \(I(S)\) is an ideal of \(C(X)\), that \[ I(S)=I(\overline S), \] and that \(S_1\subseteq S_2\) implies \(I(S_2)\subseteq I(S_1)\).
2. If \(X\) is completely regular and \(p\in X\), show that \(I(\{p\})\) is a maximal ideal of \(C(X)\).
3. If \(X\) is compact Hausdorff and \(I\) is a maximal ideal of \(C(X)\), show that \[ I=I(\{p\}) \] for some \(p\in X\).