LRU Algorithm :- Replace the least recently used page if frames are not empty.
$$\overset{\text{frames}}{\begin{array}{|c|}\hline
\textbf{f1}& \\\hline
\textbf{f2}& \\ \hline
\textbf{f3}& \\ \hline
\end{array}} \qquad \overset{\text{1}}{\begin{array}{|c|}\hline
\textbf{1}_L& \\\hline
\textbf{ }& \\ \hline
\textbf{ }& \\ \hline
\end{array}}\overset{\text{2}}{\begin{array}{|c|}\hline
\textbf{1}_L& \\\hline
\textbf{2}& \\ \hline
\textbf{ }& \\ \hline
\end{array}}\overset{\text{1}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{2}_L& \\ \hline
\textbf{ }& \\ \hline
\end{array}}\overset{\text{3}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{2}_L& \\ \hline
\textbf{3}& \\ \hline
\end{array}}\overset{\text{7}}{\begin{array}{|c|}\hline
\textbf{1}_L& \\\hline
\textbf{7}& \\ \hline
\textbf{3}& \\ \hline
\end{array}}\overset{\text{4}}{\begin{array}{|c|}\hline
\textbf{4}& \\\hline
\textbf{7}& \\ \hline
\textbf{3}_L& \\ \hline
\end{array}}\overset{\text{5}}{\begin{array}{|c|}\hline
\textbf{4}& \\\hline
\textbf{7}_L& \\ \hline
\textbf{5}& \\ \hline
\end{array}}\overset{\text{6}}{\begin{array}{|c|}\hline
\textbf{4}_L& \\\hline
\textbf{6}& \\ \hline
\textbf{5}& \\ \hline
\end{array}}\overset{\text{3}}{\begin{array}{|c|}\hline
\textbf{3}& \\\hline
\textbf{6}& \\ \hline
\textbf{5}_L& \\ \hline
\end{array}}\overset{\text{1}}{\begin{array}{|c|}\hline
\textbf{3}& \\\hline
\textbf{6}_L& \\ \hline
\textbf{1}& \\ \hline
\end{array}}$$
$X_L$ denotes the frame to be replaced.
$\therefore$ Total page faults $= 9$
Optimal Page Replacement Algorithm :- Replace the page which will not be used for longest duration of time if the frames are not empty.
$$\overset{\text{frames}}{\begin{array}{|c|}\hline
\textbf{f1}& \\\hline
\textbf{f2}& \\ \hline
\textbf{f3}& \\ \hline
\end{array}} \qquad \overset{\text{1}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{ }& \\ \hline
\textbf{ }& \\ \hline
\end{array}}\overset{\text{2}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{2}& \\ \hline
\textbf{ }& \\ \hline
\end{array}}\overset{\text{1}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{2}& \\ \hline
\textbf{ }& \\ \hline
\end{array}}\overset{\text{3}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{2}_R& \\ \hline
\textbf{3}& \\ \hline
\end{array}}\overset{\text{7}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{7}_R& \\ \hline
\textbf{3}& \\ \hline
\end{array}}\overset{\text{4}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{4}_R& \\ \hline
\textbf{3}& \\ \hline
\end{array}}\overset{\text{5}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{5}_R& \\ \hline
\textbf{3}& \\ \hline
\end{array}}\overset{\text{6}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{6}_R& \\ \hline
\textbf{3}& \\ \hline
\end{array}}\overset{\text{3}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{6}_R& \\ \hline
\textbf{3}& \\ \hline
\end{array}}\overset{\text{1}}{\begin{array}{|c|}\hline
\textbf{1}& \\\hline
\textbf{6}_R& \\ \hline
\textbf{3}& \\ \hline
\end{array}}$$
$X_R$ denotes the frame to be replaced.
$\therefore$ Total page faults $= 7$
$\therefore$ Difference in LRU and Optimal Page Replacement page faults
$= (9 - 7)$
$=2$
$\therefore$ Option $A.$ is the correct answer.
