1. ## Differentiation

Show that there exists a differentiable function $\displaystyle f:\mathbb{R}\rightarrow\mathbb{R}$ such that $\displaystyle (f(x))^5+f(x)+x=0$ $\displaystyle \forall x \in \mathbb{R}$.
I have no idea with this question. I have tried all sorts of functions but nothing works.

2. Originally Posted by worc3247
Show that there exists a differentiable function $\displaystyle f:\mathbb{R}\rightarrow\mathbb{R}$ such that $\displaystyle (f(x))^5+f(x)+x=0$ $\displaystyle \forall x \in \mathbb{R}$.
I have no idea with this question. I have tried all sorts of functions but nothing works.
This is an existence proof and we may not be able to write down the function explicitly.

We need to use the implicit function theorem and consider the function

$\displaystyle g(x,y)=x+y^5+y=0$ its Jacobian is equal to its gradient to we get

$\displaystyle \nabla g =\begin{bmatrix} 1 & y^4+1\end{bmatrix} \ne \mathbf{0}$ for all $\displaystyle (x,y)$

So by the implicit function theorem such a function must exists.