#### Answer

The value of the expression is $5$.

#### Work Step by Step

For $f\left( x \right)={{x}^{2}}$ , we can rewrite the expression as below:
$\frac{f\left( {{x}_{2}} \right)-f\left( {{x}_{1}} \right)}{{{x}_{2}}-{{x}_{1}}}=\frac{x_{2}^{2}-x_{1}^{2}}{{{x}_{2}}-{{x}_{1}}}$
Substitute ${{x}_{1}}=1\ \text{ and }\ {{x}_{2}}=4$ to get
$\begin{align}
& \frac{f\left( {{x}_{2}} \right)-f\left( {{x}_{1}} \right)}{{{x}_{2}}-{{x}_{1}}}=\frac{{{\left( 4 \right)}^{2}}-{{\left( 1 \right)}^{2}}}{4-1} \\
& =\frac{16-1}{3} \\
& =\frac{15}{3} \\
& =5
\end{align}$
Hence, the value of the expression is $\frac{f\left( {{x}_{2}} \right)-f\left( {{x}_{1}} \right)}{{{x}_{2}}-{{x}_{1}}}=5$.