$M_x$ maximal ideal in $C(X)$

Question

let $X$ be a compact Hausdorff space.consider commutative Banach algebra $C(X)$.

let for $x\in X$ is fixed. define$$M_x=\{f:f\in C(X)\ \text{and}\ f(x)=0\}$$.

then $M_x$ is proper two sided ideal. but is $M_x$ maximal ideal in $C(X)$

any hint

Answer 1

If $M$ is an ideal and $M_x\varsubsetneq M$, then there's a $f\in M$ such that $f(0)\neq0$. Now, note that$$1=\frac{f(0)}{f(0)}=\frac f{f(0)}-\frac{f-f(0)}{f(0)}\in M.$$

Answer 2

$e_x: C(X) \to \mathbb{R}$ is defined by $e_x(f) =f(x)$. Then $e_x$ is a epimorphism of unitary rings. $M_x = \ker(e_x)$ and so $C(X) / M_x \simeq \mathbb{R}$ which is a field. So $M_x$ is maximal (modding out gives a field).