Final answer:
To determine the maximum concentration of the antibiotic, the function C(t) must be differentiated and analyzed for critical points which may represent the peak concentration within the first 12 hours.
Step-by-step explanation:
To find the maximum concentration of the antibiotic during the first 12 hours after administration, we need to analyze the function C(t) = 9(e^{-0.4t} - e^{-0.6t}). To find the maximum value, we must determine the critical points of the function by taking the first derivative and setting it equal to zero, which may require using properties of exponential functions and logarithms to solve. It is also helpful to consider the behavior of the function as t approaches infinity to verify that any critical points found correspond to a maximum and not a minimum or inflection point.
The graph mentioned, showing how the plasma concentration of a drug changes over time after different forms of administration, hints that the function given most likely represents oral or intramuscular administration rather than intravenous, due to the presence of the two exponential terms in the equation and the fact that it will take time to reach the peak concentration.