Answer
$f(0)$ and $f(1)$ have opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[0,1]$.
Work Step by Step
We are given: $f(x)=3x^3-x-1$
which is a polynomial and hence a continuous function.
The Intermediate Value Theorem states that when a function is continuous on an interval $[p,q]$ and takes on values $f(p)$ and $f(q)$ at the endpoints, then the function takes on all values between $f(p)$ and $f(q)$ at some point of the interval.
We will evaluate the function at the endpoints $[0,1]$.
$f(0)=3(0)^3-(0)-1=-1$
$f(1)=3(1)^3-1-1=1$
This shows that $f(0)$ and $f(1)$ have opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[0,1]$.