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Math Help - differentiable function

  1. #1
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    differentiable function

    Show that there exists a differentiable function f: R to R such that
    (f(x))^5+f(x)+x=0 for all x in R.

    I'm thinking about using its inverse function g, but not sure if it works. Could anyone please give me some hints? Any help is appreciated!
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  2. #2
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    First off, the problem says that, for every real number x, the 5-th degree polynomial y^5 + y + x has a real zero. Do you believe this?

    You're looking for a function defined by f(x) = y, even if you can't write down a formula. I don't know how much background to assume here. You said something about an "inverse function." I would recommend considering the function g(x,y) = y^5 + y + x and applying the implicit function theorem.
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  3. #3
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    Last edited by raulk; June 8th 2010 at 10:30 AM.
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