Answer
$f(-3)=-42$ and $f(-2)=5$
This shows that $f(-3)$ and $f(-2)$ attain opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[-3,-2]$.
Work Step by Step
We are given that $f(x)=3x^3-10x +9$
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 $[-3,-2]$.
$f(-3)=3(-3)^3-10(-3) +9=-42$
$f(-2)=3(-2)^3-10(-2) +9=5$
This shows that $f(-3)$ and $f(-2)$ attain opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[-3,-2]$.
