Answer
See below.
Work Step by Step
The Intermediate Value Theorem says that if a continuous (polynomials are always continuous) function on an interval [a,b] takes values $f(a)$ and $f(b)$ at the endpoints, then the function takes all values between $f(a)$ and $f(b)$ at some point of the interval.
Evaluate the function at the endpoints.
$f(0)=3\cdot0^3-0-1=0-0-1=-1.$
$f(1)=3\cdot1^3-1-1=3-1-1=1.$
Since $-1\lt0\lt1$, according to the Intermediate Value Theorem, there must be a $0$ in the given interval.