Question

A mathematical biologist created a model to administer medicine reaction (measured in change of blood pressure or temperature) with the model given by 

2

R= m^2 (C/2- M/3)

where c is a positive constant and m is the amount of medicine absorbed into the blood. The sensitivity to the medication is defined to be the rate of change of reaction R with respect to the amount of medicine absorbed in the blood.

Answer 1

$$R=m^2\big(\dfrac{c}{2}-\dfrac{m}{3}\big)$$

(a)

$$R'=\big(m^2\big(\dfrac{c}{2}-\dfrac{m}{3}\big)\big)'=cm-m^2$$

(b)

$$R'=0=>cm-m^2=0$$

$$m=0\ or\ m=c, c>0$$

If $0<m<c,$ then $R'>0, R$ increases.

If $m>c,$ then $R'<0, R$ decreases.

The reaction is maximum at $m=c$

$$R(c)=c^2\big(\dfrac{c}{2}-\dfrac{c}{3}\big)=\dfrac{c^3}{6}$$

(c)

$$(R')'=(cm-m^2)'=c-2m$$

The instantaneous rate of change of sensitivity is the second derivative of the reaction

$$(R')'=c-2m$$

(d)

$$(R')'=0=>c-2m=0=>m=\dfrac{c}{2}$$

If $0<m<\dfrac{c}{2},$ then $(R')'>0, R'$ increases.

If $m>\dfrac{c}{2},$ then $(R')'<0, R'$ decreases.

The sensitivity is maximum at $m=\dfrac{c}{2}$

$$R'(\dfrac{c}{2})=c(\dfrac{c}{2})-(\dfrac{c}{2})^2=\dfrac{c^2}{4}$$