Let $M_{0}:=\{f \in C[0,1] \mid f(0)=0 \}$. Show that $M_0$ is not a finitely generated ideal of $C[0,1]$.

Question

We let $M_0=(f_1,f_2, \ldots ,f_n)$. Now if we consider the function $g(x)=\cos(x) \not \in M_0$, we observe that since $M_0$ is a maximal ideal of $C[0,1]$, $C[0,1]=(f_1,f_2, \ldots,f_n,g(x))$. I wanted to show that this is not possible, but I am stuck.

I saw the proof using the function $f(x)=\sum_{i=1}^n (|f_i(x)|)^{\frac{1}{2}}$, so I don't want to see that proof again. Any hints on how to proceed or any other way without actually finding a function would be helpful. Thanks in advance.