Min max: optimal quantity of medicine

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Homework Statement


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Homework Equations


Minimum/Maximum occurs when the first derivative=0

The Attempt at a Solution


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

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  • #2
SammyS
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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.
upload_2018-9-19_15-45-13.png


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} \,.##

.
 

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  • #3
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$$R'=2D\left( \frac{C}{2}-\frac{D}{3} \right)-\frac{1}{3}D^2,~~R''=C-2D,~~R''=0:~D=\frac{C}{2}$$
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
 
  • #4
SammyS
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$$R'=2D\left( \frac{C}{2}-\frac{D}{3} \right)-\frac{1}{3}D^2,~~R''=C-2D,~~R''=0:~D=\frac{C}{2}$$
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) .
 
  • #5
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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
 
  • #6
verty
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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.
 
  • #7
Ray Vickson
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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)##.
 
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Thanks
 

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