Question

optimization: problem 1
(1 point)
for some positive constant ( c ), a patients temperature change, ( t ), due to a dose, ( d ), of a drug is given by ( t=left(\frac{c}{2}-\frac{d}{3}
ight) d^{2} ).
what dosage maximizes the temperature change?
( d=)
the sensitivity of the body to the drug is defined as ( d t / d d ). what dosage maximizes sensitivity?
( d=)
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Explanation:

Step1: Expand the function \(T\)

First, expand \(T = (\frac{C}{2}-\frac{D}{3})D^{2}=\frac{C}{2}D^{2}-\frac{1}{3}D^{3}\).

Step2: Find the first - derivative of \(T\) with respect to \(D\)

Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(T^\prime=\frac{dT}{dD}=C D - D^{2}=D(C - D)\).

Step3: Find the critical points for \(T\)

Set \(T^\prime = 0\), so \(D(C - D)=0\). The solutions are \(D = 0\) or \(D=C\).

Step4: Use the second - derivative test for \(T\)

Find the second - derivative \(T^{\prime\prime}=C-2D\).
When \(D = 0\), \(T^{\prime\prime}=C>0\) (function \(T\) has a local minimum at \(D = 0\)).
When \(D = C\), \(T^{\prime\prime}=C-2C=-C<0\) (function \(T\) has a local maximum at \(D = C\)).

Step5: Find the sensitivity function \(S=\frac{dT}{dD}=CD - D^{2}\)

Step6: Find the first - derivative of \(S\) with respect to \(D\)

Let \(S = CD - D^{2}\), then \(S^\prime=C - 2D\).

Step7: Find the critical point for \(S\)

Set \(S^\prime=0\), then \(C - 2D=0\), and \(D=\frac{C}{2}\).

Step8: Use the second - derivative test for \(S\)

The second - derivative \(S^{\prime\prime}=-2<0\). So \(S\) has a local maximum at \(D=\frac{C}{2}\).

Answer:

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