Answer
By the intermediate value theorem the function $f$ has a real zero between integers 2 and 3.
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 point ‘c’ such that $g\left( c \right)=0$.
The value of the function $f$ at 2 and 3 is:
$\begin{align}
& f\left( 2 \right)=3{{\left( 2 \right)}^{3}}-8{{\left( 2 \right)}^{2}}+\left( 2 \right)+2 \\
& =24-32+2+2 \\
& =-4
\end{align}$
$\begin{align}
& f\left( 3 \right)=3{{\left( 3 \right)}^{3}}-8{{\left( 3 \right)}^{2}}+\left( 3 \right)+2 \\
& =81-72+3+2 \\
& =14
\end{align}$
Since $f\left( 2 \right)=-4$ and $f\left( 3 \right)=14$ have opposite signs, by the intermediate value theorem the function $f$ has a real zero between integers 2 and 3.
