i have no idea how to do this question. can someone help

[angelsuresh]

Suppose that antibiotics are injected into a patient to treat a sinus infection. The antibiotics circulate in the blood, slowly diffusing into the sinus cavity while simultaneously being filtered out of the blood by the liver. In a future chapter we will derive a model for the concentration of the antibiotic in the sinus cavity as a function of time since the injection:c(t) =

e−αt − e−βt

β − α

where β > α > 0.

(a) At what time does c have its maximum value?
t =




(b) At what time does the inflection point occur?
t =




What is the significance of the inflection point for the concentration function?This is the time at which the rate of change of concentration in the sinus cavity begins to ---Select--- increase decrease .

(c) Sketch the graph of c.

 

[Deleted member 4993]

Suppose that antibiotics are injected into a patient to treat a sinus infection. The antibiotics circulate in the blood, slowly diffusing into the sinus cavity while simultaneously being filtered out of the blood by the liver. In a future chapter we will derive a model for the concentration of the antibiotic in the sinus cavity as a function of time since the injection:c(t) =

e−αt − e−βt

β − α

where β > α > 0.

(a) At what time does c have its maximum value?
t =




(b) At what time does the inflection point occur?
t =




What is the significance of the inflection point for the concentration function?This is the time at which the rate of change of concentration in the sinus cavity begins to ---Select--- increase decrease .

(c) Sketch the graph of c.

What are your thoughts?

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[stapel]

...a model for the concentration of the antibiotic in the sinus cavity as a function of time since the injection [is]

\(\displaystyle c(t)\, =\, \dfrac{e^{-\alpha t}\, -\, e^{-\beta t}}{\beta\, -\, \alpha}, \mbox{ where }\, \beta\, >\, \alpha\, >\, 0\)

(a) At what time does c have its maximum value?

Max/min values come from the first derivative. What did you get? What did you find when you set the derivative equal to zero?

(b) At what time does the inflection point occur?

Inflection points come from the second derivative. What did you get? What did you find when you set the derivative equal to zero?

What is the significance of the inflection point for the concentration function? This is the time at which the rate of change of concentration in the sinus cavity begins to increase / decrease [select one]

What did you find? What are your thoughts?

(c) Sketch the graph of c.

When you thought about how medication behaves in a body and what this function is meant to model, what graph(s) did you immediately cross out? Looking at the scales and thinking about "2t" compared with "t", what did you conclude?

Please be complete. Thank you!