# How do you find on what time interval is the concentration of the drug increasing if suppose a certain drug is administered to a patient, with the percent of concentration in the bloodstream t hr later given by K(t)= 8t / (t^2 + 1)?

Apr 15, 2018

The concentration is increasing on 0 ≤ t < 1

#### Explanation:

You will need to start by differentiating.

$K ' \left(t\right) = \frac{8 \left({t}^{2} + 1\right) - 8 t \left(2 t\right)}{{t}^{2} + 1} ^ 2$

$K ' \left(t\right) = \frac{8 {t}^{2} + 8 - 16 {t}^{2}}{{t}^{2} + 1} ^ 2$

$K ' \left(t\right) = \frac{8 - 8 {t}^{2}}{{t}^{2} + 1}$

This will have a critical number when $K ' \left(t\right) = 0$.

$0 = 8 - 8 {t}^{2}$

$8 {t}^{2} = 8$

$t = \pm 1$

But since $t > 0$, only $t = 1$ is acceptable.

You will notice that whenever $t > 1$, the derivative turns negative, therefore the amount of drug in the patients system is increasing for the first hour, with 0 ≤ t < 1.

Hopefully this helps!