Concavity and The Second Derivative Test

Question

Drug Concentration: The percentage of a certain drug in the bloodstream t hours after the drug is administered is given by...

k(t) = 3t/t^2+4

For example, after 1 hour the concentration is given by...

k(1) = 3(1)/(1)^2+4 = 3/5% = 0.6% = 0.006

(a.) Find the time at which concentration is a maximum

(b.) Find the maximum concentration

Answer 1

k(t) = 3t/(t²+1)

dk/dt = ((t²+1)(3) - (3t)(2t))/(t²+1)²

Set dk/dt to 0 -3t² + 3 = 0 t = +/- 1

You can look at concavity (sign of second derivative in order to establish if max, min, or inflection.

The function goes to 0 when t goes to +/- infinity.

d²k/dt² = d/dt(dk/dt) = ((t²+1)(-6t) - (-3t²+3)2(t²+1)(2t))/(t²+1)⁴

Plugging in -1: 2nd derivative is 12 (+ curvature, smiley face: min)

Plugging in 1: 2nd derivative is -12 (- curvature, frowny face, max)

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