Final answer:
First Lady Michelle Obama uses an argument by authority in her speech for the Drink Up program, utilizing her position and role as a mother to advocate for children's health.
Step-by-step explanation:
The type of argument used by First Lady Michelle Obama in her speech promoting the Drink Up program would be best described as an argument by authority based on her position as the First Lady and a mother (option A). In her role, she leverages her authority to advocate for policies and practices that improve public health, similar to her efforts with the Let's Move! campaign. The appeal to her authority is not only derived from her official position but also from her personal role as a mother, both of which lend credibility to her argument and encourage the audience to follow the program's recommendations.