Prove that for a polynomial function $f$ with degree $n$ if $f(x) \geq 0$ for all $x$, then $f(x) + f'(x) + \cdots + f^{(n)} (x) \geq 0$.
Give me some hints for this and please explain to me how you have come to those hints.
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2I believe you should specify that $n=\deg f$, since it is not true otherwise. – L. F. Dec 09 '13 at 11:20
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2you can see this http://math.stackexchange.com/questions/397973/sum-of-polynoms-of-given-property/397997#397997 – math110 Dec 09 '13 at 11:26
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sorry guys, I just edited – le4m Dec 09 '13 at 11:28
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1Yes I see this is a duplicate. So should I delete this question? By the way, the solution there seems ingenious. How on earth can I think of it? – le4m Dec 09 '13 at 11:32
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@julypraise : you don't have o delete it, it will be closed if there are 2 more votes to close... As you said it may sound as if it is ingenious... wait for some time some one might help you... – Dec 09 '13 at 11:50
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have you tried seeing this by considering degree $1$ and degree $2$ polynomials separately... ? this may not be generalized but then i was expecting you may get some idea... – Dec 09 '13 at 11:55
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Yes, I think those simple cases may motivate one to think of the second solution on the linked page. But I never thought of use of $e^x$. – le4m Dec 09 '13 at 12:01
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@math110 How did you think of use of $e^x$ in such a way? Is it some kind of general technique in analysis, in measure theory, etc..? – le4m Dec 09 '13 at 22:15