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Math Help - Differentiable Function Question

  1. #1
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    Differentiable Function Question

    Show that there exists a differentiable function f : R → R such that
    (f(x))^5 + f(x) + x = 0 for all x ∈ R.
    [Hint: If f exists and has an inverse function g what equation must g satisfy? ]

    I haven't got a clue how to start this one. Any help would be greatly appreciated.

    Thanks
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  2. #2
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    Quote Originally Posted by DeFacto View Post
    Show that there exists a differentiable function f : R → R such that
    (f(x))^5 + f(x) + x = 0 for all x ∈ R.
    [Hint: If f exists and has an inverse function g what equation must g satisfy? ]
    I think you'll find this easier if you write y = f(x). Then y^5+y+x=0. The hint suggests that you should consider the inverse function, in other words that you should look at x as a function of y. Then x = -y^5-y, and \tfrac{dx}{dy} = -5y^4-1<0 for all y. Now use the inverse function theorem to conclude that y is a differentiable function of x.
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