Take 5 minutes to prepare this exercise.
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A detailed solution is then proposed to you.
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Question
For any natural integer
greater or equal to
we define the application
from
to
by:
Prove that equation
has a unique solution in interval
Study the variations of function
then the sign of its values in
and in
Use then the bijection's theorem.
Application
is differentiable, hence continuous, by the generic theorems.Let us sutdy the variations of application
hence
This implies that application
is strictly negative on interval
therefore that application
is strictly decreasing on
and
hence application
defines a bijection from
to
which implies that it takes once and only once the zero value.
