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!
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!
First off, the problem says that, for every real number $\displaystyle x$, the 5-th degree polynomial $\displaystyle y^5 + y + x$ has a real zero. Do you believe this?
You're looking for a function defined by $\displaystyle f(x) = y$, 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 $\displaystyle g(x,y) = y^5 + y + x$ and applying the implicit function theorem.