1. ## 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

2. Originally Posted by DeFacto
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.