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Thread: Intermediate Value Theorem

  1. #1
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    Intermediate Value Theorem

    The problem:
    If $\displaystyle a$ and $\displaystyle b$ are positive numbers, prove that the equation
    $\displaystyle \frac{a}{x^3+2x-1}+\frac{b}{x^3+x-2}=0$
    has at least one solution in the interval (-1,1).
    I was hoping for some pointers on how to proceed.
    Sorry, I made a mistake earlier that $\displaystyle 2x$ was supposed to be a $\displaystyle 2x^2$.
    Last edited by Dotdash13; Oct 5th 2010 at 09:05 PM.
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  2. #2
    Member Mollier's Avatar
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    Hi,

    say we write f(x) = $\displaystyle \frac{a}{x^3+2x-1}+\frac{b}{x^3+x-2}$.
    Then as $\displaystyle x \rightarrow -1, f(x) \rightarrow -\frac{a}{4}-\frac{b}{4}$.
    Similarly you can show that as $\displaystyle x$ approaches $\displaystyle 1$ the function value is positive.
    Since $\displaystyle a$ and $\displaystyle b$ are positive numbers, the function $\displaystyle f $ "crosses the x-axis" in the interval $\displaystyle (-1,1)$, and so a solution exists.

    I'm sure you can write this in more mathy terms.

    Hope this helps!
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