Let $r$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I = [a,b]$ and vanishing at $n + 1$ distinct points of $I$. Show that there exists $c \in I$ such that $r ^ { ( n ) } ( c ) = 0$.
Let $r$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I = [a,b]$ and vanishing at $n + 1$ distinct points of $I$. Show that there exists $c \in I$ such that $r ^ { ( n ) } ( c ) = 0$.