Answer
By the intermediate value theorem the function $f$ has a real zero between integers −3 and −2.
Work Step by Step
According to the intermediate value theorem if $g\left( a \right)$ and $g\left( b \right)$ are of different signs, that is $g\left( a \right)g\left( b \right)<0$ , then there will exist at least one a point ‘c’ such that $g\left( c \right)=0$.
So, to the value of the function $f$ at −3 and −2 is:
$\begin{align}
& f\left( -3 \right)=3{{\left( -3 \right)}^{3}}-10\left( -3 \right)+9 \\
& =-81+30+9 \\
& =-42
\end{align}$
$\begin{align}
& f\left( -2 \right)=3{{\left( -2 \right)}^{3}}-10\left( -2 \right)+9 \\
& =-24+20+9 \\
& =5
\end{align}$
Since, $f\left( -3 \right)=-42$ and $f\left( -2 \right)=5$ have opposite signs, by the intermediate value theorem the function $f$ has a real zero between integers −3 and −2.
