Question

use the intermediate value theorem to show that there is a positive number whose 5th power is exactly 1 more than itself.

Answer 1

Let $f(x)=x^5-(x+1)$. Then $f(1)=-1$ and $f(2)=29$. By the intermediate value theorem, it follows that there is some $c\in(1,2)$ such that $f(c)\in[f(-1),f(2)]=[-1,29]$.

This guarantees that there is some $c$ between -1 and 2 such that $f(c)=0$, i.e. there is some $c$ such that $c^5=c+1$.