Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+…+a_{1}x=0$ has positive roots, then...












-1














Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x=0$ has positive roots, then $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+ a_{1}=0$ has positive roots.
P/s: My grammar isn't good, so that my question is hard to understand. I’m sorry for the inconvenience






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  • Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    – Martin R
    Nov 25 at 13:50








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    This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
    – Batominovski
    Nov 25 at 13:51












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    – xbh
    Nov 25 at 13:51
















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Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x=0$ has positive roots, then $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+ a_{1}=0$ has positive roots.
P/s: My grammar isn't good, so that my question is hard to understand. I’m sorry for the inconvenience






derivatives







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  • Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    – Martin R
    Nov 25 at 13:50








  • 1




    This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
    – Batominovski
    Nov 25 at 13:51












  • Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
    – xbh
    Nov 25 at 13:51














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1





Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x=0$ has positive roots, then $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+ a_{1}=0$ has positive roots.
P/s: My grammar isn't good, so that my question is hard to understand. I’m sorry for the inconvenience






derivatives







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Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x=0$ has positive roots, then $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+ a_{1}=0$ has positive roots.
P/s: My grammar isn't good, so that my question is hard to understand. I’m sorry for the inconvenience






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derivatives






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asked Nov 25 at 13:46









Trần Tuấn

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  • Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    – Martin R
    Nov 25 at 13:50








  • 1




    This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
    – Batominovski
    Nov 25 at 13:51












  • Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
    – xbh
    Nov 25 at 13:51


















  • Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    – Martin R
    Nov 25 at 13:50








  • 1




    This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
    – Batominovski
    Nov 25 at 13:51












  • Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
    – xbh
    Nov 25 at 13:51
















Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– Martin R
Nov 25 at 13:50






Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– Martin R
Nov 25 at 13:50






1




1




This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
– Batominovski
Nov 25 at 13:51






This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
– Batominovski
Nov 25 at 13:51














Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
– xbh
Nov 25 at 13:51




Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
– xbh
Nov 25 at 13:51










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Let $alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $mathbb R$)
$$Longrightarrow exists cin(0,alpha):f'(c) = 0$$
which is the required statement to be proven.



Hope it is helpful






share|cite|improve this answer





















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    Let $alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $mathbb R$)
    $$Longrightarrow exists cin(0,alpha):f'(c) = 0$$
    which is the required statement to be proven.



    Hope it is helpful






    share|cite|improve this answer


























      1














      Let $alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $mathbb R$)
      $$Longrightarrow exists cin(0,alpha):f'(c) = 0$$
      which is the required statement to be proven.



      Hope it is helpful






      share|cite|improve this answer
























        1












        1








        1






        Let $alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $mathbb R$)
        $$Longrightarrow exists cin(0,alpha):f'(c) = 0$$
        which is the required statement to be proven.



        Hope it is helpful






        share|cite|improve this answer












        Let $alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $mathbb R$)
        $$Longrightarrow exists cin(0,alpha):f'(c) = 0$$
        which is the required statement to be proven.



        Hope it is helpful







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 25 at 13:56









        Martund

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