Inflection Points

Question

A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S(t) = At^pe^−kt

 (where t > 0) is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A = 0.02, p = 4, k = 0.07, and t is measured in minutes, estimate the times t corresponding to the inflection points. (Round your answers to two decimal places.)

t = .... min (smaller value)

t = .... min (larger value)

Answer 1

An inflection points occur when the second derivative of a function passes through the x-axis. aka when the second derivative changes signs.

given S(t) = 0.02 * t^4 * e^(-0.07*t)

dS/dt = e^(-0.07*t) * t^3 * (0.08 - 0.0014*t)

d^2S/dt^2 = e^(-0.07*t) * t^2 * (0.000098 * t^2 - 0.0112*t + 0.24)

d^2S/dt^2 is equal to zero when t = 0, 200/7, 600/7

no sign change at t = 0, so inflection points at

t = 200/7, 600/7

Answer 2

same answer as Chengun J.,,,,most likely correct

set second derivative S''(t)=0

using just symbols for S''(t) setting to zero, dividing out Ae^-kt

and the letting p=4

find want

k²t⁴-8kt²+12t²=0,,,,divide out t² and have quadratic

t=(4/k)+/-(2/k)

inflection points at t=28.5 and 85.5 minutes approximately.