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
We denote by
a polynomial.
Prove that equation
has only a finite number of roots.
Apply successively several times Rolle's theorem, until the polynomial disappears.
Let
be the degree of the polynomial. We set
the number of roots of equation
is equal to the number of zeros of application
Let us assume that application
has
zeros, from the result of the previous exercise, this implies that application
takes the zero value at least once.
However the derivative of order
of a polynomial of degree
is zero, hence
which is never zero.
The equation has therefore at most
solutions.