After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function
C(t) = 3(e−0.4t − e−0.6t)
where the time t is measured in hours and C is measured in µg/mL. What is the maximum concentration of the antibiotic during the first 12 hours? (Round your answer to four decimal places.)

To find the maximum concentration of the antibiotic during the first 12 hours, we need to determine the critical points of the concentration function \( C(t) = 3(e^{-0.4t} - e^{-0.6t}) \) by following these steps:

1. **Differentiate the function** with respect to \( t \) to find the critical points.
2. **Set the derivative equal to zero** to find the values of \( t \) where the concentration is at a maximum or minimum.
3. **Evaluate the second derivative** (optional, to confirm whether it's a maximum).
4. **Evaluate \( C(t) \) at the critical points and at \( t = 0 \) and \( t = 12 \) to find the maximum value** within the interval.

Let’s start by differentiating the function and solving for critical points.The critical point occurs at \( t \approx 2.0273 \) hours.

Next, I will evaluate the concentration function \( C(t) \) at this critical point, as well as at the endpoints \( t = 0 \) and \( t = 12 \), to find the maximum concentration during the first 12 hours.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, you can attempt evaluating the function at these points manually:

1. The critical point \( t \approx 2.0273 \)
2. At \( t = 0 \)
3. At \( t = 12 \)

This will help you find the maximum concentration over the interval.

Do you have any other questions? 

Here are some related questions you might want to explore:
1. How does the rate of concentration change over time?
2. What would happen if the time interval was extended beyond 12 hours?
3. Can you find the second derivative to confirm whether the critical point is a maximum?
4. How would you model this if the antibiotic decayed differently?
5. Can you solve this problem using numerical methods?

**Tip:** Checking endpoints is crucial when determining maximum values over a closed interval.

After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function C(t) = 3(e−0.4t − e−0.6t) where the time t is measured in hours and C is measured in µg/mL. What is the maximum concentration of the antibiotic during the first 12 hours? (Round your answer to four decimal places.)

Learn how to find the maximum concentration of an antibiotic in the bloodstream using calculus. This problem involves differentiation, exponential functions, and identifying critical points to determine the maximum concentration within the first 12 hours. Suitable for Grades 11-12 or college-level students.

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