Optimize Real World Solution: Reaction to Medicine

Question

The reaction of the body to a dose of medicine can sometimes be represented by an equation of the form R=M^2[(C/2) - (M/3)] where C is a positive constant and M is the amount of medicine absorbed in the blood. If the reaction is a change in blood​ pressure, R is measured in millimeters of mercury. If the reaction is a change in temperature, R is measured in​ degrees, and so on.

Find the amount of medicine to which the body is most sensitive by finding the value of M that maximizes the derivative dR/dM.

Use an expression with C as a variable.

Answer 1

Rewrite R = M^2 * C/2 - M^3 / 3, now differentiating with respect to M we find

dR/dM = CM - M^2, denote this function by f(M). The goal is to maximize the function f. So we start by finding its critical points, for that we need to differentiate in M again

f'(M) = C - 2M, equating to zero we see that the only critical point of f is M=C/2. Further we note that if M>C/2 then f'(M)<0, that is f decreases after C/2 and similarly it increases before C/2.

Thus, M=C/2 is the point where f(M) or equivalently dR/dM attains its maximum value.