Editing Rolle's theorem (section)

== Generalization to higher derivatives ==

We can also generalize Rolle's theorem by requiring that {{mvar|f}} has more points with equal values and greater regularity.  Specifically, suppose that
* the function {{mvar|f}} is {{math|''n'' − 1}} times [[Smoothness#Differentiability_classes|continuously differentiable]] on the closed interval {{closed-closed|''a'', ''b''}} and the {{mvar|n}}th derivative exists on the open interval {{open-open|''a'', ''b''}}, and
* there are {{mvar|n}} intervals given by {{math|''a''<sub>1</sub> < ''b''<sub>1</sub> ≤ ''a''<sub>2</sub> < ''b''<sub>2</sub> ≤ ⋯ ≤ ''a<sub>n</sub>'' < ''b<sub>n</sub>''}} in {{closed-closed|''a'', ''b''}} such that {{math|1=''f&thinsp;''(''a<sub>k</sub>'') = ''f&thinsp;''(''b<sub>k</sub>'')}} for every {{mvar|k}} from 1 to {{mvar|n}}. 
Then there is a number {{mvar|c}} in {{open-open|''a'', ''b''}} such that the {{mvar|n}}th derivative of {{mvar|f}} at {{mvar|c}} is zero.
[[File:Rolle Generale.svg|thumb|290x290px|The red curve is the graph of function with 3 roots in the interval {{closed-closed|−3, 2}}. Thus its second derivative (graphed in green) also has a root in the same interval.]]

The requirements concerning the {{mvar|n}}th derivative of {{mvar|f}} can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand limits defined above with {{math|''f&thinsp;''{{isup|(''n'' − 1)}}}} in place of {{mvar|f}}.

Particularly, this version of the theorem asserts that if a function differentiable enough times has {{mvar|n}} roots (so they have the same value, that is 0), then there is an internal point where {{math|''f&thinsp;''{{isup|(''n'' − 1)}}}} vanishes.

===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.