Min max: optimal quantity of medicine

[Karol]

Homework Statement

Homework Equations

Minimum/Maximum occurs when the first derivative=0

The Attempt at a Solution

$$R'=2D\left( \frac{C}{2}-\frac{D}{3} \right)-\frac{1}{3}D^2$$
$$R'=0~\rightarrow~D=C$$

 

[SammyS]

Homework Statement

View attachment 230882

Homework Equations

Minimum/Maximum occurs when the first derivative=0

The Attempt at a Solution

$$R'=2D\left( \frac{C}{2}-\frac{D}{3} \right)-\frac{1}{3}D^2$$
$$R'=0~\rightarrow~D=C$$

Read carefully.

It discusses the point where $\ R^\prime (D) \$ is a maximum, not about the max of $R(D)\,.$

Also, it may help to write R(D) as: $\ \displaystyle R(D) = \frac{D^2C}{2}-\frac{D^3}{3} \,.$

.

 

[Karol]

$$R'=2D\left( \frac{C}{2}-\frac{D}{3} \right)-\frac{1}{3}D^2,~~R''=C-2D,~~R''=0:~D=\frac{C}{2}$$
But the greatest change in R for a small change in D is where R has a maximum, hence where R'=0, not where R''=0

 

[SammyS]

$$R'=2D\left( \frac{C}{2}-\frac{D}{3} \right)-\frac{1}{3}D^2,~~R''=C-2D,~~R''=0:~D=\frac{C}{2}$$
But the greatest change in R for a small change in D is where R has a maximum, hence where R'=0, not where R''=0

Read the problem again. It's R'(D) which you need to find the maximum for, not finding the maximum for R(D) .

 

[Karol]

Yes, that's correct, i need the maximum for R', but why?
At the point where R has a maximum, i think, a small change in D makes a big change in R

 

[verty]

Yes, that's correct, i need the maximum for R', but why?
At the point where R has a maximum, i think, a small change in D makes a big change in R

No, that's where R' is a maximum. R' = 0 is where there is no change with a change in D.

 

[Ray Vickson]

Yes, that's correct, i need the maximum for R', but why?
At the point where R has a maximum, i think, a small change in D makes a big change in R

No: at the maximum a small change in $D$ makes NO change in $R$! The tangent line to the graph $R = f(D)$ is horizontal at an interior maximum; that is why we look for points where the derivative vanishes when maximizing or minimizing.

You really need to have a better intuitive understanding of this material, and to help with that I suggest that you substitute some numerical value for $C$, then plot $R(D)$ and $R'(D)$.

 

[Karol]

Thanks