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A closed rectangular box has volume 32 cm$^3$.

What is the maximum volume under the elliplical paraboloid?Without Extreme Value Theorem, how do we find absolute extrema?Prove an inequality by Second partial derivative test?A smooth function $f(x)$ has a unique local and global minimum. What happens to its location as $f(x)$ varies smoothly in time?Show that the critical point of this function is a global minimum (geometric median)Finding the global maxima and minima on closed intervalIsosceles Triangle/CircleFinding Minimum of Error Function. Confused About Meaning of Critical Values.Two variable function on an unbounded setShow that global minima exists or not

1

$begingroup$

Find the lengths of the edges giving the minimum surface area of following the three steps below:

Step1: Let the length, width, and height of the box be $x,y,z$. Write $z$ in terms of x and y using the condition that the volume of the box is 32cm$^3$.

 I have xyz=32cm^3 which then gives me 32/xy=z

Step2: Write the surface $S$ as a function of $x,y,z$, then replace z with the expression in Step1 to write $S$ as a function of $x$ and $y$.

 I don't know what to do here

Step3: Find the critical point(s) of the function $S(x,y)$ in step 2 and determine the local minimum.

 This will be the partial derivatives of the function in step2

Step4: Is the local minimum the global minimum? Make your conclusion on the values of $x,y$ ans $z$ that minimize the surface area.

 I don't know what to do here

I also don't know what bounds I should be operating in here

partial-derivative maxima-minima

share|cite|improve this question

asked Mar 17 at 23:38

NeedHelpNeedHelp

243

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I believe it's asking for surface area. Can you determine what the surface area should be?
  $endgroup$
  – WaveX
  Mar 17 at 23:43
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  (That is, as a function of $x,y,z$?)
  $endgroup$
  – Gerry Myerson
  Mar 17 at 23:46
  

  
  
  
  

add a comment | 

1

$begingroup$

Find the lengths of the edges giving the minimum surface area of following the three steps below:

Step1: Let the length, width, and height of the box be $x,y,z$. Write $z$ in terms of x and y using the condition that the volume of the box is 32cm$^3$.

 I have xyz=32cm^3 which then gives me 32/xy=z

Step2: Write the surface $S$ as a function of $x,y,z$, then replace z with the expression in Step1 to write $S$ as a function of $x$ and $y$.

 I don't know what to do here

Step3: Find the critical point(s) of the function $S(x,y)$ in step 2 and determine the local minimum.

 This will be the partial derivatives of the function in step2

Step4: Is the local minimum the global minimum? Make your conclusion on the values of $x,y$ ans $z$ that minimize the surface area.

 I don't know what to do here

I also don't know what bounds I should be operating in here

partial-derivative maxima-minima

share|cite|improve this question

asked Mar 17 at 23:38

NeedHelpNeedHelp

243

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I believe it's asking for surface area. Can you determine what the surface area should be?
  $endgroup$
  – WaveX
  Mar 17 at 23:43
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  (That is, as a function of $x,y,z$?)
  $endgroup$
  – Gerry Myerson
  Mar 17 at 23:46
  

  
  
  
  

add a comment | 

$begingroup$

Find the lengths of the edges giving the minimum surface area of following the three steps below:

Step1: Let the length, width, and height of the box be $x,y,z$. Write $z$ in terms of x and y using the condition that the volume of the box is 32cm$^3$.

 I have xyz=32cm^3 which then gives me 32/xy=z

Step2: Write the surface $S$ as a function of $x,y,z$, then replace z with the expression in Step1 to write $S$ as a function of $x$ and $y$.

 I don't know what to do here

Step3: Find the critical point(s) of the function $S(x,y)$ in step 2 and determine the local minimum.

 This will be the partial derivatives of the function in step2

Step4: Is the local minimum the global minimum? Make your conclusion on the values of $x,y$ ans $z$ that minimize the surface area.

 I don't know what to do here

I also don't know what bounds I should be operating in here

partial-derivative maxima-minima

share|cite|improve this question

asked Mar 17 at 23:38

NeedHelpNeedHelp

243

$endgroup$

 I have xyz=32cm^3 which then gives me 32/xy=z

 I don't know what to do here

 This will be the partial derivatives of the function in step2

 I don't know what to do here

share|cite|improve this question

asked Mar 17 at 23:38

NeedHelpNeedHelp

243

share|cite|improve this question

asked Mar 17 at 23:38

NeedHelpNeedHelp

243

asked Mar 17 at 23:38

NeedHelpNeedHelp

243

asked Mar 17 at 23:38

NeedHelpNeedHelp

243

1 Answer
1

active

oldest

votes

1

$begingroup$

The surface of your box has two $x times y$ rectangles, two $x times z$ rectangles, and two $y times z$ rectangles, so $S=2(xy+xz+yz)$

share|cite|improve this answer

answered Mar 18 at 0:00

Ross MillikanRoss Millikan

300k24200375

$endgroup$

add a comment | 

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1

$begingroup$

The surface of your box has two $x times y$ rectangles, two $x times z$ rectangles, and two $y times z$ rectangles, so $S=2(xy+xz+yz)$

share|cite|improve this answer

answered Mar 18 at 0:00

Ross MillikanRoss Millikan

300k24200375

$endgroup$

add a comment | 

1

$begingroup$

The surface of your box has two $x times y$ rectangles, two $x times z$ rectangles, and two $y times z$ rectangles, so $S=2(xy+xz+yz)$

share|cite|improve this answer

answered Mar 18 at 0:00

Ross MillikanRoss Millikan

300k24200375

$endgroup$

add a comment | 

$begingroup$

The surface of your box has two $x times y$ rectangles, two $x times z$ rectangles, and two $y times z$ rectangles, so $S=2(xy+xz+yz)$

share|cite|improve this answer

answered Mar 18 at 0:00

Ross MillikanRoss Millikan

300k24200375

$endgroup$

share|cite|improve this answer

answered Mar 18 at 0:00

Ross MillikanRoss Millikan

300k24200375

answered Mar 18 at 0:00

Ross MillikanRoss Millikan

300k24200375

answered Mar 18 at 0:00

Ross MillikanRoss Millikan

300k24200375