# generalized Rolle's theorem trouble

• Nov 30th 2009, 03:21 PM
dannyboycurtis
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 \$\displaystyle f^{(n)}(c)=0\$
• Nov 30th 2009, 05:31 PM
Jose27
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.