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 point ‘c’ such that $g\left( c \right)=0$.
So, the value of the function $f$ at −3 and −2 will be:
$\begin{align}
& f\left( -3 \right)={{\left( -3 \right)}^{3}}+{{\left( -3 \right)}^{2}}-2\left( -3 \right)+1 \\
& =-27+9+6+1 \\
& =-11
\end{align}$
$\begin{align}
& f\left( -2 \right)={{\left( -2 \right)}^{3}}+{{\left( -2 \right)}^{2}}-2\left( -2 \right)+1 \\
& =-8+4+4+1 \\
& =1
\end{align}$
Since $f\left( -3 \right)=-11$ and $f\left( -2 \right)=1$ have opposite signs, by the intermediate value theorem the function $f$ has a real zero between integers −3 and −2.
