TITLE: A closed rectangular box has volume 32 cm$^3$.
QUESTION [1 upvotes]: 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

REPLY [1 votes]: The surface of your box has two $x \times y$ rectangles, two $x \times z$ rectangles, and two $y \times z$ rectangles, so $S=2(xy+xz+yz)$
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