[14 points] Let $\mathbb { R } _ { + }$ denote the set of positive real numbers. A one-to-one and onto function $f : \mathbb { R } _ { + } \rightarrow \mathbb { R } _ { + }$ is called golden if $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R } _ { + }$. (i) Find all golden functions (if any) of the form $f ( x ) = a x ^ { b }$. Find all golden functions (if any) of the form $f ( x ) = a b ^ { x }$. In both cases $a$ and $b$ are suitable real numbers. (ii) Show that there is no one-to-one and onto function $f : \mathbb { R } \rightarrow \mathbb { R }$ such that $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R }$.
[14 points] Let $\mathbb { R } _ { + }$ denote the set of positive real numbers. A one-to-one and onto function $f : \mathbb { R } _ { + } \rightarrow \mathbb { R } _ { + }$ is called golden if $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R } _ { + }$.\\
(i) Find all golden functions (if any) of the form $f ( x ) = a x ^ { b }$. Find all golden functions (if any) of the form $f ( x ) = a b ^ { x }$. In both cases $a$ and $b$ are suitable real numbers.\\
(ii) Show that there is no one-to-one and onto function $f : \mathbb { R } \rightarrow \mathbb { R }$ such that $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R }$.