R=M^2(c/2-m/3)
dR/dM=CM-M^2
I found the derivative. Now how would I find the vale of M that maximize the derivative dR/dM?
set it to zero, and solve for m. You get two solutions. Use the second derivative to see which one is the max.
I get M=C. How do I go from there? What do I do?
M=C is one of the solutions for a max/min. Use the second derivative to see which. M=0 is the other.
how can you tell which is the max/min? They are both variables. c=positive constanct, m=amt of medicine absorbed in the blood, dR/dM=sensitivity of the body of medicine
How do you get m=0? from taking the second derivative? does c=1?
Can you check my work on how I get the derivative of R=M^2(c/2-m/3) NEXT R=1/2(CM^2)-1/3(M^3) NEXT dR/dM=CM-M^2. I checked in the back, and that was the answer for the first part of the equation. Isn't the derivative of constant c, zero? I am confused.
C is a constant so you use the product rule.
2m(c/2-m/3)+m^2(1/3)
2m being the derivative of m^2. times the original second term. then the derivative of the second term times the original first.
1/3 being the derivative of the (c/2-m/3) because C is a constant
To find the values of M that maximize the derivative dR/dM, you need to set the derivative equal to zero and solve for M. In this case, you have dR/dM = CM - M^2.
Setting this equation to zero:
0 = CM - M^2
Rearranging the equation:
M^2 = CM
Now, you can solve for the value of M. To do this, divide both sides of the equation by C:
M^2/C = M
This simplifies to:
M = 0 or M = C
These are the two solutions you mentioned. M = 0 and M = C.
To determine which one is the maximum, you need to use the second derivative test. Taking the second derivative of R with respect to M, you get:
d^2R/dM^2 = C - 2M
If you plug in M = 0, the second derivative becomes:
d^2R/dM^2 = C - 2(0) = C
If you plug in M = C, the second derivative becomes:
d^2R/dM^2 = C - 2C = -C
Since C is a positive constant, C > 0, and its negative value -C would be negative. This means that when M = C, the second derivative is negative, indicating a maximum.
Therefore, the value M = C maximizes the derivative dR/dM of the function R.
To find the value of M that maximizes the derivative dR/dM, you first correctly set dR/dM equal to zero:
dR/dM = CM - M^2 = 0.
Now, to solve for M, you can rearrange the equation:
CM = M^2.
Dividing both sides by M:
C = M.
So, you correctly found that M = C. However, there is also another solution:
M = 0.
To determine whether M = C or M = 0 corresponds to a maximum or minimum, you need to use the second derivative test.
First, take the second derivative of R with respect to M:
d^2R/dM^2 = d/dM (CM - M^2) = C - 2M.
Now, substitute M = C and M = 0 into the second derivative:
When M = C: d^2R/dM^2 = C - 2C = -C < 0.
When M = 0: d^2R/dM^2 = C - 2(0) = C > 0.
Since the second derivative is negative when M = C and positive when M = 0, the value M = C corresponds to a maximum, while M = 0 corresponds to a minimum.
So, the value of M that maximizes the derivative dR/dM is M = C.