# Math Help - generalized Rolle's theorem trouble

1. ## generalized Rolle's theorem trouble

Hi. I am having trouble proving the following:
Let f be continuous on [a,b], n time differentiable on (a,b), and f(x_0)=f(x_1)=...=f(x_n)=0 for some x_0,x_1,...,x_n in [a,b]. Show that there exists c in (a,b) such that $f^{(n)}(c)=0$

2. Notice that the case $n=1$ is simply Rolle's theorem. Now applying Rolle to each interval of the form $[x_i,x_{i+1}]$ with $i=0,...,n$ we have that there exist $n$ points, say $y_0,...,y_{n-1}$ where the first derivative is $0$, and applying Rolle again in $[y_i,y_{i+1}]$ we get $n-1$ points where $f^{(2)}$ is $0$. Arguing inductively we get that there exists $n-(j-1)$ points where $f^{(j)}$ is zero so we get 2 points where $f^{(n-1)}$ is zero and applying Rolle one more time the result follows.