Editing Rolle's theorem (section)

===Proof===
The proof uses [[mathematical induction]]. The case {{math|1=''n'' = 1}} is simply the standard version of Rolle's theorem. For {{math|''n'' > 1}}, take as the induction hypothesis that the generalization is true for {{math|''n'' − 1}}. We want to prove it for {{mvar|n}}. Assume the function {{mvar|f}} satisfies the hypotheses of the theorem. By the standard version of Rolle's theorem, for every integer {{mvar|k}} from 1 to {{mvar|n}}, there exists a {{mvar|c<sub>k</sub>}} in the open interval {{open-open|''a<sub>k</sub>'', ''b<sub>k</sub>''}} such that {{math|1=''f&thinsp;''′(''c<sub>k</sub>'') = 0}}. Hence, the first derivative satisfies the assumptions on the {{math|''n'' − 1}} closed intervals {{math|[''c''<sub>1</sub>, ''c''<sub>2</sub>], …, [''c''<sub>''n'' − 1</sub>, ''c<sub>n</sub>'']}}. By the induction hypothesis, there is a {{mvar|c}} such that the {{math|(''n'' − 1)}}st derivative of {{math|''f&thinsp;''′}} at {{mvar|c}} is zero.