The reaction R(x) of a patient to a drug dose size of x depends on the type of drug. For a certain drug, it was determined that a good description of the relationship is :R(x)= Ax^2(B-x) on domain [0,B]
where A and B are positive constants . The sensitivity of the patient's body to the drug is defined to be R'(X)
a) for what value of x is the reaction a max and what is the max reaction value
b) for what value of x is the sensitivity a maximum? what is that maximum sensitivity?
a) Ah, the good old maximum reaction value! Let's find it, shall we? To find the maximum reaction value, we need to find the critical points of the function R(x). The critical points are where the derivative R'(x) is equal to zero.
So, first, let's find R'(x). Using the power rule and the product rule of differentiation, we find:
R'(x) = 2Ax(B - x) - Ax^2
Setting R'(x) equal to zero and solving for x:
0 = 2Ax(B - x) - Ax^2
Expanding and rearranging a bit:
Ax^2 - 2ABx + 2Ax = 0
Factoring out an x:
x(Ax - 2AB + 2A) = 0
Now, we have two possibilities:
1) x = 0
2) Ax - 2AB + 2A = 0
Since we are dealing with the domain [0, B], we can discard the first possibility (x = 0), as it's not within the given domain.
Now, let's solve for x in the second equation:
Ax - 2AB + 2A = 0
Ax = 2AB - 2A
Dividing both sides by A:
x = 2B - 2
So, the value of x where the reaction is a maximum is x = 2B - 2. To find the maximum reaction value, substitute this value of x into the original R(x) equation:
R(max) = A(2B - 2)^2(B - (2B - 2))
b) Now, let's move on to the maximum sensitivity! The sensitivity of the patient's body to the drug is defined as R'(x). Therefore, we need to find the maximum value of R'(x), which will give us the maximum sensitivity.
We already have R'(x) from the previous part:
R'(x) = 2Ax(B - x) - Ax^2
To find the maximum value of R'(x), we need to find the critical points of R'(x). So, set R''(x) (the second derivative of R(x)) equal to zero:
R''(x) = 2A(B - 2x) - 2Ax
0 = 2A(B - 2x) - 2Ax
0 = 2AB - 4Ax - 2Ax
0 = -4Ax + 2AB
4Ax = 2AB
2Ax = AB
x = B/2
So, the value of x where the sensitivity is a maximum is x = B/2. To find the maximum sensitivity, substitute this value of x into the R'(x) equation:
R'(max) = 2A(B - B/2) - A(B/2)^2
a) To find the value of x at which the reaction is a maximum, we need to find the critical points of the function R(x). The critical points occur when the derivative, R'(x), equals zero.
First, let's find the derivative of R(x) with respect to x:
R(x) = Ax^2(B - x)
R'(x) = 2Ax(B - x) + Ax^2(-1)
= 2ABx - 2Ax^2 - Ax^2
= 2ABx - 3Ax^2
To find the critical points, we set R'(x) equal to zero and solve for x:
2ABx - 3Ax^2 = 0
Factor out an x:
x(2AB - 3Ax) = 0
This equation will be true when either x = 0 or 2AB - 3Ax = 0.
For x = 0, R'(x) = 2AB(0) - 3A(0)^2 = 0. Therefore, x = 0 is a critical point.
For 2AB - 3Ax = 0:
2AB = 3Ax
x = (2AB)/(3A)
x = 2B/3
Since the domain of R(x) is [0, B], we need to check if x = 0 and x = 2B/3 are within this range. Since 0 is within the range, we only consider x = 2B/3.
Therefore, the value of x at which the reaction R(x) is a maximum is x = 2B/3. To find the maximum reaction value, we substitute x = 2B/3 into R(x):
R(x) = A(2B/3)^2(B - 2B/3)
Simplifying further,
R(x) = (4AB^2/9)(B - 2B/3)
= (4/9)AB^2(B - 2B/3)
= (4/9)AB^2(3B - 2B)
= (4/9)(B^3)
Therefore, the maximum reaction value is (4/9)(B^3).
b) To find the value of x at which the sensitivity is a maximum, we need to find the critical points of the sensitivity function R'(x). The sensitivity is given by the derivative of the reaction function R(x).
R'(x) = 2AB - 3Ax
To find the critical points, we set R'(x) equal to zero and solve for x:
2AB - 3Ax = 0
Factor out an x:
x(2AB - 3A) = 0
This equation will be true when either x = 0 or 2AB - 3A = 0.
For x = 0, R'(x) = 2AB - 3A(0) = 2AB. Since A and B are positive constants, this cannot equal zero. Therefore, x = 0 is not a critical point.
For 2AB - 3A = 0:
2AB = 3A
B = 3/2
Therefore, the value of x at which the sensitivity R'(x) is a maximum is x = B = 3/2, and the maximum sensitivity is R'(3/2) = 2AB - 3A(3/2) = AB - 9A/2.
To find the value of x at which the reaction is a maximum (part a), we need to find the critical points of the function R(x) = Ax^2(B-x) and then determine the maximum of those critical points.
1. Find the derivative of R(x) with respect to x:
R'(x) = 2Ax(B - x) + Ax^2(-1)
= 2ABx - 2Ax^2 - Ax^2
= -3Ax^2 + 2ABx
2. Set R'(x) equal to zero and solve for x to find the critical points:
-3Ax^2 + 2ABx = 0
x(-3Ax + 2AB) = 0
From this equation, we have two possibilities:
a) x = 0
b) -3Ax + 2AB = 0
3. Find the value of x when -3Ax + 2AB = 0:
-3Ax + 2AB = 0
-3Ax = -2AB
x = 2A/B
Now, let's determine the maximum reaction value (R(x)) at the critical points.
4. Evaluate R(x) at the critical points:
a) When x = 0:
R(0) = A(0)^2(B-0)
= 0
b) When x = 2A/B:
R(2A/B) = A(2A/B)^2(B - 2A/B)
= (4A^3/B^2)(B^2 - 2AB + 4A^2)
= 4A^3(B^2 - 2AB + 4A^2) / B^2
We can simplify this expression further, but the final value depends on the specific values of A and B.
To find the value of x at which the sensitivity is a maximum (part b), we need to find the critical points of the derivative R'(x) and then determine the maximum of those critical points.
1. Find the derivative of R'(x) with respect to x:
R''(x) = -6Ax + 2AB
2. Set R''(x) equal to zero and solve for x to find the critical points:
-6Ax + 2AB = 0
-6Ax = -2AB
x = 1/3B
Now, let's determine the maximum sensitivity (R'(x)) at the critical point.
3. Evaluate R'(x) at the critical point:
R'(1/3B) = -3A(1/3B)^2 + 2AB(1/3B)
= -A/3B + 2A/3
= A(2 - 1/3B) / 3
Again, the final value depends on the specific values of A and B.
R' = 2Ax(B-x) - Ax^2 = Ax(2B-3x)
(a) R'=0 when x=0,2B/3
R" = A(2B-3x) - 3Ax = 2A(B-3x)
(b) R"=0 when x=B/3