Chapter 2 - Limits - 2.8 Intermediate Value Theorem - Exercises - Page 86: 1

See the proof below.

Since the function $ f(x)=x^3+x $ is continuous on $[1,2]$ and $ f(1)=2\neq f(2)=10$ and $9$ is between $ f(1)$ and $f(2)$, then by the Intermediate Value Theorem the function $ f(x)$ takes on the value $9$ for some $ x\in (1,2)$.