Final answer:
Mr. Jenkins shows signs of mastering the developmental task of integrity versus despair, feeling fulfillment from his life accomplishments and experiences.
Step-by-step explanation:
Mr. Jenkins, at 82 years of age, reflecting on his life's accomplishments and finding comfort and fulfillment in his experiences, is showing signs of mastering the developmental task known as integrity versus despair. This task is part of Erik Erikson's theory of psychosocial development, which suggests that in the final stage of life, people reflect on their achievements and experiences to either feel a sense of satisfaction, which is integrity, or a sense of failure, which leads to despair. Since Mr. Jenkins is feeling at peace and fulfilled, he has successfully achieved a sense of integrity.