I used both agglomerative clustering and k-means on a dataset and see the results below. Result from agglomerative clustering was demonstrated with silhouette score while kmeans with inertia score. The inertia score was pretty smooth overall, while the highest silhouette score was at 2 clusters. I wonder people think would be optimal number of clusters and why? I know for silhouette scores usually the one with the highest value would be the optimal number of clusters but I rarely see results of two clusters so is two clusters even meaningful?
My somewhat heterodox opinion is that a cluster analysis is meaningful if it has meaning and that this is a substantive question rather than a statistical one.
Sure, you can use the various statistical criteria for number of clusters as a sort of guideline, but these criteria often disagree with each other.
So ... look at the various cluster solutions. Which one tells you something interesting, something surprising, something useful? That's the one to go with.