This is my first post here (I hope that this hasn't been asked / answered before). First let me state the Generalized Rolle's Theorem as it is presented to me, then I'll ask my question.
(Generalized Rolle's Theorem) Assume that $f \in C[a, b]$ and that $f'(x), f''(x), \dotsc, f^{(n)}(x)$ exist over $(a, b)$ and $x_0, x_1, \dotsc, x_n \in [a, b]$. If $f(x_j) = 0$ for $j = 0, 1, \dotsc, n$, then there exists a number $c$, with $c \in (a, b)$, such that $f^{(n)}(c) = 0$.
I was thinking about the simple example $f(x) = x$, where $x \in [0, 1]$. If we let $x_0 = 0 \in [0, 1]$, then the conditions for the Generalized Rolle's Theorem are satisfied, but there is no number $c \in (0, 1)$ such that $f^{(0)}(c) = f(c) = 0$. What am I doing wrong? Is the answer that $n \geq 1$? Shouldn't they state this?