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
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
Notice that the case is simply Rolle's theorem. Now applying Rolle to each interval of the form with we have that there exist points, say where the first derivative is , and applying Rolle again in we get points where is . Arguing inductively we get that there exists points where is zero so we get 2 points where is zero and applying Rolle one more time the result follows.