The concentration of a drug in a patient's bloodstream t hours after it is taken after it is taken is given by

C(t) = 0.016t/(t+2)^2 mg/cm^3
Find the maximum concentration of the drug and the time at which it occurs.

The maximum concentration of the drug occurs when the derivative of C(t) is equal to 0.

C'(t) = 0.016(2t+2)/(t+2)^3

0 = 0.016(2t+2)/(t+2)^3

2t+2 = 0

t = -1

The maximum concentration of the drug occurs at t = -1 and is 0.016 mg/cm^3.

To find the maximum concentration of the drug and the time at which it occurs, we need to find the derivative of the concentration function and then solve for when the derivative equals zero.

Step 1: Find the derivative of the concentration function C(t)
To find the derivative of C(t), we can use the quotient rule. The quotient rule states that if we have a function of the form f(t)/g(t), then the derivative is given by [f'(t)g(t) - f(t)g'(t)] / [g(t)]^2.

Using this rule, we can find the derivative of C(t):
C'(t) = [ (0.016(t+2)^2 - 2(0.016t)(t+2))/(t+2)^4 ] mg/cm^3

Step 2: Find the critical points by solving C'(t) = 0
Setting C'(t) = 0, we can solve for t:
[ (0.016(t+2)^2 - 2(0.016t)(t+2))/(t+2)^4 ] = 0

Simplifying the equation:
0.016(t+2)^2 - 2(0.016t)(t+2) = 0

Expand and rearrange the equation:
0.016t^2 + 0.064t + 0.064 - 0.032t^2 - 0.064t = 0

Combine like terms:
-0.016t^2 = -0.064

Divide both sides by -0.016 to isolate t^2:
t^2 = 4

Take the square root of both sides:
t = +- 2

So the critical points are t = 2 and t = -2.

Step 3: Determine the maximum concentration
To determine whether these points correspond to a maximum or minimum, we can check the concavity of the function. Since the question asks for the maximum concentration, we need to find the maximum point.

To check concavity, we can take the second derivative. If the second derivative is positive, then the function is concave up and the critical point corresponds to a minimum. If the second derivative is negative, then the function is concave down and the critical point corresponds to a maximum.

Step 4: Find the second derivative
To find the second derivative, we take the derivative of the first derivative, C'(t):
C''(t) = [ (0.016(t+2)^2 - 2(0.016t)(t+2))/(t+2)^4 ]' = [-0.032/(t+2)^3] mg/cm^3

Step 5: Evaluate C''(t) at t = 2
Plug in t = 2 into C''(t):
C''(2) = -0.032/(2+2)^3 = -0.032/64 = -0.0005 mg/cm^3

Since the second derivative is negative, the function is concave down, which means that the critical point t = 2 corresponds to a maximum concentration.

So the maximum concentration of the drug occurs at t = 2 hours.

To find the maximum concentration of the drug and the time at which it occurs, we need to determine the critical points of the concentration function by finding its derivative and setting it equal to zero. Let's start by finding the derivative of C(t).

1. Find the derivative of C(t) with respect to t.
To find the derivative, we can use the quotient rule:

C'(t) = [ (t+2)^2 * (0.016) - 0.016t * 2(t+2) ] / (t+2)^4
= [ 0.032(t+2) - 0.032t ] / (t+2)^4
= [ 0.032t + 0.064 - 0.032t ] / (t+2)^4
= 0.064 / (t+2)^4

Now we can find the critical points by setting C'(t) equal to zero:

0.064 / (t+2)^4 = 0

2. Solve for t.
Multiplying both sides by (t+2)^4, we get:
0.064 = 0

Since the equation simplifies to 0 = 0, it means there are no values of t that make C'(t) equal to zero. Therefore, there are no critical points.

3. Check the endpoints.
To ensure that the maximum concentration occurs within the given time range, we need to check the endpoints of the domain. In this case, we have:

t ≥ 0 (because we can't have a negative time)

4. Calculate C(t) at the endpoints.
Let's evaluate C(t) at the endpoints of the given time range:

a) t = 0:
C(0) = 0.016 * 0 / (0+2)^2
= 0

b) t approaches infinity:
As t approaches infinity, the concentration approaches:
C(t) = 0.016t/(t+2)^2
= 0.016/t

5. Determine the maximum concentration and time.
Based on the calculations, we can observe that the concentration at t = 0 is 0, and as t approaches infinity, the concentration approaches 0. Therefore, the maximum concentration of the drug is 0 mg/cm^3, and it occurs at t = 0 hours.

In summary, the maximum concentration of the drug is 0 mg/cm^3, and it occurs at t = 0 hours.