differentiable function

• Jun 8th 2010, 04:09 AM
nngktr
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!
• Jun 8th 2010, 08:57 AM
drjerry
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.
• Jun 8th 2010, 10:19 AM
raulk
.