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Math Help - Intermediate Value Theorem

  1. #1
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    Intermediate Value Theorem

    The problem:
    If a and b are positive numbers, prove that the equation
    \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 2x was supposed to be a 2x^2.
    Last edited by Dotdash13; October 5th 2010 at 10:05 PM.
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  2. #2
    Member Mollier's Avatar
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    Hi,

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