How to show the following for infinitely differentiable function. [closed]

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Let $f$ be a real valued infinitely differentiable function with domain $mathbb{R}$. Assume $f(0) = 0$ and $f(1) = 1$. Also $f(x)geq 0$ for all $x$. Show that there exists a positive integer $n$ and a real number $x$ such that $$f^{(n)}(x) < 0$$










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closed as off-topic by caverac, Davide Giraudo, José Carlos Santos, KReiser, RRL Dec 9 '18 at 3:56


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Davide Giraudo, José Carlos Santos, KReiser, RRL

If this question can be reworded to fit the rules in the help center, please edit the question.













  • $begingroup$
    This is Problem A5 of this year's (2018) Putnam competition. No discussions are supposed to be made public until the exam has ended!
    $endgroup$
    – mlerma54
    Dec 1 '18 at 18:43
















2












$begingroup$


Let $f$ be a real valued infinitely differentiable function with domain $mathbb{R}$. Assume $f(0) = 0$ and $f(1) = 1$. Also $f(x)geq 0$ for all $x$. Show that there exists a positive integer $n$ and a real number $x$ such that $$f^{(n)}(x) < 0$$










share|cite|improve this question









$endgroup$



closed as off-topic by caverac, Davide Giraudo, José Carlos Santos, KReiser, RRL Dec 9 '18 at 3:56


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Davide Giraudo, José Carlos Santos, KReiser, RRL

If this question can be reworded to fit the rules in the help center, please edit the question.













  • $begingroup$
    This is Problem A5 of this year's (2018) Putnam competition. No discussions are supposed to be made public until the exam has ended!
    $endgroup$
    – mlerma54
    Dec 1 '18 at 18:43














2












2








2


2



$begingroup$


Let $f$ be a real valued infinitely differentiable function with domain $mathbb{R}$. Assume $f(0) = 0$ and $f(1) = 1$. Also $f(x)geq 0$ for all $x$. Show that there exists a positive integer $n$ and a real number $x$ such that $$f^{(n)}(x) < 0$$










share|cite|improve this question









$endgroup$




Let $f$ be a real valued infinitely differentiable function with domain $mathbb{R}$. Assume $f(0) = 0$ and $f(1) = 1$. Also $f(x)geq 0$ for all $x$. Show that there exists a positive integer $n$ and a real number $x$ such that $$f^{(n)}(x) < 0$$







real-analysis functions derivatives






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asked Dec 1 '18 at 16:32









Mittal GMittal G

1,193515




1,193515




closed as off-topic by caverac, Davide Giraudo, José Carlos Santos, KReiser, RRL Dec 9 '18 at 3:56


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Davide Giraudo, José Carlos Santos, KReiser, RRL

If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by caverac, Davide Giraudo, José Carlos Santos, KReiser, RRL Dec 9 '18 at 3:56


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Davide Giraudo, José Carlos Santos, KReiser, RRL

If this question can be reworded to fit the rules in the help center, please edit the question.












  • $begingroup$
    This is Problem A5 of this year's (2018) Putnam competition. No discussions are supposed to be made public until the exam has ended!
    $endgroup$
    – mlerma54
    Dec 1 '18 at 18:43


















  • $begingroup$
    This is Problem A5 of this year's (2018) Putnam competition. No discussions are supposed to be made public until the exam has ended!
    $endgroup$
    – mlerma54
    Dec 1 '18 at 18:43
















$begingroup$
This is Problem A5 of this year's (2018) Putnam competition. No discussions are supposed to be made public until the exam has ended!
$endgroup$
– mlerma54
Dec 1 '18 at 18:43




$begingroup$
This is Problem A5 of this year's (2018) Putnam competition. No discussions are supposed to be made public until the exam has ended!
$endgroup$
– mlerma54
Dec 1 '18 at 18:43










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