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`.