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Math Help - find how many solutions this ln polynomial equation has..

  1. #1
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    find how many solutions this ln polynomial equation has..

    <br />
2x^3lnx+x^2-4x+a=0<br />
    i thought of building a taylor series around 0 for ln
    but ln(0) is undefined

    ??
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let 0,\infty)\to\mathbf{R}" alt="f(x)=2x^3\ln x+x^2-4x+a, \ f0,\infty)\to\mathbf{R}" />

    We use the Rolle's sequence.

    f'(x)=6x^2\ln x+2x^2+2x-4

    f'(x)=0\Rightarrow x=1 and f' is increasing so it has an unique solution.

    \lim_{x\searrow 0}f(x)=a, \ f(1)=a-3, \ \lim_{x\to\infty}f(x)=\infty

    If a\in(-\infty,0] then the Rolle's sequence is - \ - \ + and f has a solution x_1\in(1,\infty)

    If a\in(0,3) then the Rolle's sequence is + \ - \ + and f has two solutions x_1\in(0,1), \ x_2\in(1,\infty)

    If a=3 then the Rolle's sequence is + \ 0 \ + and f has the solution x=1

    If a>3 then the Rolle's sequence is + \ + \ + and f has no real solution.
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