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Thread: Mimimum value of a certain polynomial

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    Senior Member bkarpuz's Avatar
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    Mimimum value of a certain polynomial

    Dear all,

    I need to calculate the minimum value of the following poynomial:
    $\displaystyle P_{n}(\lambda):=\lambda(\lambda-1)\cdots(\lambda-n+1),$
    where $\displaystyle n\in\mathbb{N}$ is even and $\displaystyle p\in\mathbb{R}$.
    I can transform this polynomial into
    $\displaystyle P_{n}(\lambda):=\lambda^{n}+\sum_{i=0}^{n-1}a_{i+1}\lambda^{n-1-i},$
    by the means of Vieta's formula (PlanetMath: Vieta's formula) but this time it will not be easy to find the roots of $\displaystyle P_{n}^{\prime}=0$.
    I really wonder if this is something well-known.

    Thanks for the help again.

    Note: It also helps me to find good upper and lower bounds for $\displaystyle \min\nolimits_{\lambda\in[0,(n-1)/2]}\big\{P_{n}(\lambda)\big\}$.
    Last edited by bkarpuz; Feb 26th 2009 at 02:59 AM. Reason: Note is inserted
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