Answer
$f(-5)=-58$ and $f(-4)=2$
This shows that $f(-5)$ and $f(-4)$ attain opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[-5,-4]$.
Work Step by Step
We are given that $f(x)=2x^3+6x^2-8x+2$
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 $[-5,-4]$.
$f(-5)=2(-5)^3+6(-5)^2-8(-5)+2=-58$
$f(-4)=2(-4)^3+6(-4)^2-8(-4)+2=2$
This shows that $f(-5)$ and $f(-4)$ attain opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[-5,-4]$.
