Question

for some positive constant c, a patients temperature change, t, due to a dose, d, of a drug is given by ( t = (\frac{c}{2}-\frac{d}{3})d^{2} ).
what dosage maximizes the temperature change?
( d=square )
the sensitivity of the body to the drug is defined as ( dt/dd ). what dosage maximizes sensitivity?
( d=square )

Explanation:

Step1: Expand the function

Expand \(T = (\frac{C}{2}-\frac{D}{3})D^{2}\) to \(T=\frac{C}{2}D^{2}-\frac{1}{3}D^{3}\).

Step2: Find the first - derivative for temperature change

Differentiate \(T\) with respect to \(D\). Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we get \(T^\prime=\frac{dT}{dD}=CD - D^{2}=D(C - D)\).
Set \(T^\prime = 0\). Then \(D(C - D)=0\), which gives \(D = 0\) or \(D = C\).
Differentiate \(T^\prime\) with respect to \(D\) to get \(T^{\prime\prime}=C-2D\).
When \(D = 0\), \(T^{\prime\prime}=C>0\) (function is concave - down). When \(D = C\), \(T^{\prime\prime}=C - 2C=-C<0\). So the temperature change is maximized at \(D = C\).

Step3: Find the sensitivity function

The sensitivity \(S=\frac{dT}{dD}=CD - D^{2}\).
Differentiate \(S\) with respect to \(D\). Using the power rule, \(S^\prime=C - 2D\).
Set \(S^\prime = 0\). Then \(C-2D = 0\), which gives \(D=\frac{C}{2}\).
Differentiate \(S^\prime\) with respect to \(D\) to get \(S^{\prime\prime}=-2<0\). So the sensitivity is maximized at \(D=\frac{C}{2}\).

Answer:

For the temperature - change maximization: \(D = C\).
For the sensitivity maximization: \(D=\frac{C}{2}\).