Take 10 minutes to prepare this exercise.
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Question
We denote by
a polynomial of degree
with real coefficients and we assume that
is positive, meaning:
We set
Prove that:
Introduce the application
defined from
to
by
Determine the derivative of function
to study its variations and conclude using the limit of
when
goes to
We define a function
from
to
by
therefore:
Polynomial
is of degree
therefore
is zero, which gives, with an index translation:
We infer:
Since polynomial
has positive values,
has negative values, hence function
is decreasing.
By compared growths,
goes towards
when
goes to
therefore
which implies
