How does Kate earn Petruchio's favor when meeting Vincentio?
When Petruchio and Katherine encounter the traveling Vincentio (Act IV, Scene 5), Petruchio refers to the elderly man as a "gentlewoman" (l. 33) and asks his wife to greet the "fair lovely maid" (l. 37). Kate's response (one of the great comic moments in Shakespeare) begins, "Young budding virgin, fair and fresh and sweet..." (l. 41). If Petruchio says Vincentio is a young woman, then Kate says he is a young woman. Earlier in the same scene, Petruchio insisted that the sun was the moon. Kate argued at first; then -- realizing that she would never reach home and be fed by doing so -- she capitulated. Here, she doesn't resist him at all. When Petruchio then reprimands her for her "madness" in calling Vincentio a "maiden" (l. 48), Kate abjectly apologizes to the old man for her "mad mistaking" (l. 53). Katherine is now willing wholeheartedly to obey her husband. The "shrew" hath finally been tamed.