Rolle's Theorem
Revision as of 22:46, 14 February 2008 by Shreyas patankar (talk | contribs)
Rolle's theorem is an important theorem among the class of results regarding the value of the derivative on an interval.
Statement
Let
Let be continous on
and differentiable on
Let
Then
such that
Proof
The result is trivial for the case . Hence, let us assume that
is a non-constant function.
Let and
Without loss of generality, we can assume that
By the Extremum value theorem, such that
Assume if possible
Let
Hence, such that
i.e. ,
Thus we have that if
, contradicting the assumption that
is a maximum.
Similarly we can show that leads to contradiction.
Therefore,
QED