Answer
By the intermediate value theorem the function $f$ has a real zero between integers 1 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 $g\left( c \right)=0$.
And, the value of the function $f$ at 1 and 2 is:
$\begin{align}
& f\left( 1 \right)={{\left( 1 \right)}^{5}}-{{\left( 1 \right)}^{3}}-1 \\
& =1-1-1 \\
& =-1
\end{align}$
$\begin{align}
& f\left( 2 \right)={{\left( 2 \right)}^{5}}-{{\left( 2 \right)}^{3}}-1 \\
& =32-8-1 \\
& =23
\end{align}$
Since $f\left( 1 \right)=-1$ and $f\left( 2 \right)=23$ have opposite signs, by the intermediate value theorem the function $f$ has a real zero between integers 1 and 2.
