Suppose we have a polynomial satisfying p+p′′′>=p′+p′′ for all x. Then prove that p(x)>=0 for all x.

furajat4h 2022-09-15 Answered
Suppose we have a polynomial satisfying p + p p + p for all x. Then prove that p ( x ) 0 for all x.
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Answers (1)

acorazarxf
Answered 2022-09-16 Author has 9 answers
Define
f ( x ) = e x ( p ( x ) 2 p ( x ) + p ( x ) ) .
Then lim x f ( x ) = 0 and
f ( x ) = e x ( p ( x ) p ( x ) p ( x ) + p ( x ) ) 0 ,
i.e. f is increasing. It follows that f 0, and hence
p 2 p + p 0.
Define
g ( x ) = e x ( p ( x ) p ( x ) ) .
Then lim x + g ( x ) = 0 and
g ( x ) = e x ( p ( x ) 2 p ( x ) + p ( x ) ) 0 ,
i.e. g is decreasing. It follows that g 0, and hence
p p 0.
Define
h ( x ) = e x p ( x ) .
Then lim x + h ( x ) = 0 and
h ( x ) = g ( x ) 0 ,
i.e. h is decreasing. Therefore, either h > 0 or h 0. The conclusion follows.

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