$($c$)(10$ pts$)$ The dendrogram of a hierarchical clustering method shows the data points in the x axis with the y$-$axis showing the average $($or single or complete$)$ link distance threshold when a given cluster splits into two clusters $($going from the top to bottom$).$ One way to choose the optimal number of clusters with a dendrogram is to find the $\backslash ($ y $\backslash )-$axis region with maximum distance $($vertically$)$ which does not include a new branching $($split$).$ Then the number of clusters can be determined by the number of branches that intercept a horizontal line within that region.

i$.(5$ pts $)$ Based on the above criteria, find the appropriate number of clusters $($and the region with maximum distance $($vertically$)$ which does not include a new branching $($split$))$ in the following dendrogram $($you may directly draw on the figure below for your answer$).$

ii$.(5$ pts$)$ Explain why the above method is a reasonable way to choose the number of clusters. $($Hint: use your y$-$axis region to justify your choice$).$