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.