Question 64
a) Show that, if $n$ is a positive integer and $f: \mathbb{R} \to \mathbb{R}$ is a derivative, then the function $x \mapsto (f(x))^n$, $x \in \mathbb{R}$ has the intermediate values property.
b) Give an example of a function $f: \mathbb{R} \to \mathbb{R}$ which is a derivative, but none of the functions $x \mapsto (f(x))^n$, $x \in \mathbb{R}$, $n \in \mathbb{Z}$, $n \ge 2$, is a derivative.
Dorin Andrica and Vlad Crișan
b) Give an example of a function $f: \mathbb{R} \to \mathbb{R}$ which is a derivative, but none of the functions $x \mapsto (f(x))^n$, $x \in \mathbb{R}$, $n \in \mathbb{Z}$, $n \ge 2$, is a derivative.
Dorin Andrica and Vlad Crișan