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