Answer
$f(-1)=-6$ and $f(0)=2$
This shows that $f(-1)$ and $f(0)$ attain opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[-1,0]$.
Work Step by Step
We are given that $f(x)=x^4+8x^3-x^2+2$
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 $[-1,0]$.
$f(-1)=(-1)^4+8(-1)^3-(-1)^2+2=-6$
$f(0)=(0)^4+8(0)^3-(0)^2+2=2$
This shows that $f(-1)$ and $f(0)$ attain opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[-1,0]$.
