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Math Help - 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:
    P_{n}(\lambda):=\lambda(\lambda-1)\cdots(\lambda-n+1),
    where n\in\mathbb{N} is even and p\in\mathbb{R}.
    I can transform this polynomial into
    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 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 \min\nolimits_{\lambda\in[0,(n-1)/2]}\big\{P_{n}(\lambda)\big\}.
    Last edited by bkarpuz; February 26th 2009 at 02:59 AM. Reason: Note is inserted
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