(a) If M is any maximal ideal of $R$, then $\exists c\in[0,1]$ s.t. $M=M_C=\{f|f(c)=0\}$
So somehow I need to prove that every maximum ideal is the kernal of an evaluation at some $c \in [0,1]$....
(b) $x \neq y \to M_x \neq M_y$
This seems intuitive but I need help showing it rigorously. If $x \neq y$ then of course there will be at least one function that has a zero at $x$ but not $y$ and vice versa...
(c) $M_c \neq <x-c>$
$sin(2 \pi x/c) \in M_c$ and $sin(2 \pi x/c) \notin <x-c>$... this is the only one I think that i've got
(d) $M_c$ is not finitely generated
Perhaps I would proceed by contradiction and assume it is finitely generated by $<f_1,f_2,....,f_k>$ and then try to construct a $f \in M_c$ s.t. $f \notin f_1R+f_2R+....f_kR$
Insight greatly appreciated!!