Take 5 minutes to prepare this exercise.
Then, if you lack ideas to begin, look at the given clue and start searching for the solution.
A detailed solution is then proposed to you.
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Question
Let
be an application of class
on
We assume that application
is zero in
distinct real numbers.
Prove that the derivative of order
of
takes the zero value at least once.
Apply successively several times Rolle's theorem: the derivative application of
takes at least
times the zero value, the derivative of
at least
times.....
Let
be the
real numbers such that
From Rolle's theorem, there exist
real numbers:
such that
The real numbers
are distinct.
We have the same reasonning, there always exists:
such that
We repeat the reasonning until we obtain the existence of a real number
such that
