hello. i am having a hard time with this question. I attached an image of the problem. Handwritten answers are prefered for understanding. (a) Find the time t = tM at which c(t) has its maximum value, as a function of α and β.(b) At what time does an inflection point occur? Explain the significance of the inflection pointfor the graph of the concentration function c(t).(c) Now consider the particular parameter values C = 2, α = 0.3 and β = 0.5; that is,c(t) = 10(e^−0.3t − e^−0.5t),where time t is measured in hours, and concentration c(t) is in µg/mL.• What are the maximum and minimum concentrations of the antibiotic during the first 18hours after injection?• Use the second derivative test to show that that the internal extremum at some time tMwith 0 < tM < 12 is indeed a maximum.• Also find the time ti at which the inflection point occurs.(d) Sketch the graph of c(t) for t ∈ [0, 18]. Indicate the maximum value, minimum value andinflection point on your graph. Suppose that a patient is injected with antibiotics to treat, say, a sinus infection. The antibioticscirculate in the blood, slowly diffusing into the sinus cavity while simultaneously being filtered outof the blood by the liver.Following the injection, the antibiotic concentration in the body decays with time. Previously, wemodelled such decay using a single decaying exponential function of the form e-at; this is called asingle-compartment model, and essentially assumes a homogeneous distribution of the drug in thebody. A more realistic multi-compartment" description takes into account that the distribution ofthe drug to different parts of the body, and also elimination/metabolic breakdown, may occur atdifferent rates (for example, drug is distributed rapidly in the bloodstream and to highly perfusedorgans such as the liver and kidneys; but then diffuses to other body tissues and organs more slowly).For antibiotic treatment of a sinus infection, under certain conditions the concentration c(t) of theantibiotic in the sinus cavity, as a function of time t since the injection, is given by a function of theformBtc(t) = ce-atB-a(for given positive constants a, ß and C).where 3 > a > 0

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Answered: Suppose that a patient is injected with…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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hello. i am having a hard time with this question. I attached an image of the problem. Handwritten answers are prefered for understanding.

(a) Find the time t = tM at which c(t) has its maximum value, as a function of α and β.

(b) At what time does an inflection point occur? Explain the significance of the inflection point
for the graph of the concentration function c(t).

(c) Now consider the particular parameter values C = 2, α = 0.3 and β = 0.5; that is,
c(t) = 10(e^−0.3t − e^−0.5t),
where time t is measured in hours, and concentration c(t) is in µg/mL.
• What are the maximum and minimum concentrations of the antibiotic during the first 18
hours after injection?
• Use the second derivative test to show that that the internal extremum at some time tM
with 0 < tM < 12 is indeed a maximum.
• Also find the time ti at which the inflection point occurs.

(d) Sketch the graph of c(t) for t ∈ [0, 18]. Indicate the maximum value, minimum value and
inflection point on your graph.