Climax is a literary term referring to turning point of a story. This is also usually considered the most interesting point. After the climax, things cannot return to the way they were. This is what makes it climactic. The initial conflict is either resolved or changed.
This story centers around the disappearance of 10 year old Lord Saltire from Priory School. Holmes is called in to investigate that kidnapping at the request of the boy’s father. The investigation essentially ends when Holmes asks the Duke for the reward money, tells him where the boy is, and accuses the Duke. This is the climax because the story completely changes. The boy is no longer missing, so the search for him is over.
As it turns out, the kidnapper was the Duke’s personal secretary, who turns out to be his illegitimate son. The man he hired murdered the teacher and ran off, and Wilder confessed to the Duke, who tried to save further family embarrassment by hiding the boy.