Question

Is there a bijection \(f:\mathbb R^+\to\mathbb R^+\) s.t. \(f'(x)=f^{-1}(x)\)?

Answer 1

ill help

Answer 2

Existence can be shown because \(f(x)=\phi^{-\frac\phi{\phi+1}}x^\phi\) satisfies the conditions.

Is this unique?

Answer 3

\[f'(f(x))=x,\forall x\in I:=(0,\infty),\therefore f\in C^\infty(I)\]

Is \(f\) analytic?

Answer 4

Never mind, I was just on my way to M.SE I guess.

Answer 5

Oh wow, check this out.

http://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions

Answer 6

Never mind, I suppose I'm just talking to myself here...

Answer 7

THANKS FOR ALL THE HELP.

Just kidding, I love you.

Answer 8

lol