After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the following function

Question

C(t) = 7(e −0.5t − 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 4 hours and when does it happen? The complete argument should be provided for the full points.

Answer 1

The maximum of a continuous differentiable function like C(t)

occurs at the boundary of its domain [0,4] or at a critical point.

Critical point of a function occurs where its derivative is 0.

C'(t) = 7((-0.5)e^-0.5t − (-0.6)e^-0.6t ) = 7(-0.5e^-0.5t + 0.6e^-0.6t )

To find any critical point you set C'(t) = 0 and solve the equation

7(-0.5e^-0.5t + 0.6e^-0.6t ) = 0

0.5e^-0.5t = 0.6e^-0.6t

Multiply both sides of the equation by e^0.6t

0.5e^0.1t = 0.6

e^0.1t = 1.2

0.1t = ln(1.2)

t = 10×ln(1.2) ≈ 1.8

So the critical point occurs at approximately 1.8 hours, which is within

the domain [0,4], so it is a candidate for the sought maximum.

So we have three candidates for maximum, and we have to compare their values

C(0) = 0

C(4) = 7(e^-2 - e^-2.4) ≈ 0.31

C(10×ln(1.2)) = 7(e^-5ln(1.2) - e^-6ln(1.2) )= 7(1.2^-5 - 1.2^-6) ≈ 0.47

The maximum on the interval [0,4] occurs at approximately 1.8 hours.