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# After an antibiotic taken is taken, the concentration of the antibiotic in the bloodstream is modeled by the function$$C(t) = 8(e^{-0.4t} - e^{-0.6t})$$where the time $t$ is measured in hours and $C$ is measured in $\mu$g/mL. What is the maximum concentration of the antibiotic during the first $12$ hours?

## $C(2.027)=1.18$

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Okay. Ah This function cfx Equalling eight times E. To the negative 80.4 X minus E. To the negative 0.6 X. Represents the amount of concentration uh of an antibiotic in the bloodstream uh After x. Number of hours. And uh in the time in a role from 0 to 12 hours, we want to know when was the concentration the maximum? Now you can find a maximum point on a function by finding the first derivative of the function and setting that first derivative equal to zero. But this function uh it's a little bit easier to find a maximum point. This function simply by graphing it. So here is the graph of the function. And if we look at the interval from X equals 02 X. Equals 12. You can see that the maximum point on the function happens right here. So when the time is 2.027 hours we have the maximum amount of the antibiotic in the blood stream which is 1.185 micrograms per millimeter. Okay, so the maximum concentration in the blood stream during the 1st 12 hours is 1.185 uh micrograms per liter. Her miller miller leader

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