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: $f(x)=8x^4-4x^3-2x-1$
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 $[0,1]$:
$f(0)=8(0)^4-4(0)^3-2(0)-1=-1$
$f(1)=8(1)^4-4(1)^3-2(1)-1=1$
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 $[0,1]$.
