Take 10 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 a differentiable application from
to
such that
and
Let
be a non zero natural integer, prove that there exists
real numbers
belonging to
such that
Slice the segment
in
segments of equal length and apply the Rolle's theorem to each of them.
Let us consider the steady subdivision of step
meaning:
From the formula of mean values:
By summing the previous equalities for
from
to
we obtain:
The terms of the sum of the left member of the previous equality are zero when taken two by two, meaning there remains
which gives us the expected result using the hypotheses on
and