# Thread: Maxima and minima section

1. ## Maxima and minima section

I was assigned a homework problem in my max and min section, and one problem has me completely stooped. It doesn't even provide me a function to work with, and I looked above and below the problem with no clue. The problem is:

Find the shape of the rectangular box of volume Vzero for which the sum of the edge lengths is least.

Keep in mind that for all the other problems assigned I was using the Second Partials Test, which consisted of D = D(xo,yo) = fxx(xo,yo)*fyy(xo,yo) - f^2xy(xo,yo). Thanks in advance.

2. Originally Posted by pakman

Keep in mind that for all the other problems assigned I was using the Second Partials Test, which consisted of D = D(xo,yo) = fxx(xo,yo)*fyy(xo,yo) - f^2xy(xo,yo). Thanks in advance.
The second partial test is only used to determine is a point is a max or a min. It is not used to find absolute extrema, in that case you just make the partials equal to zero.

We know that,
$\displaystyle V_0=xyz$
And we want to maximize,
$\displaystyle S(x,y,z)=2x+2y+2z$
First you can solve for "z"
Thus,
$\displaystyle z=\frac{V_0}{xy}$
Thus,
$\displaystyle S(x,y)=2x+2y+\frac{2V_0}{xy}$
Now find,
$\displaystyle S_x=0$
$\displaystyle S_y=0$

3. So for S(x,y) did you plug back in xyz for Vo? I'm getting stuck at the partial derivative since I dont know what to do with Vo. If I plug xyz back in I'll just get 2 for Sx so it seems like I'm doing something wrong here.

4. Originally Posted by pakman
So for S(x,y) did you plug back in xyz for Vo? I'm getting stuck at the partial derivative since I dont know what to do with Vo. If I plug xyz back in I'll just get 2 for Sx so it seems like I'm doing something wrong here.
No $\displaystyle V_0$ is just a constant function. It is only a number. Pretend instead of $\displaystyle V_0$ you had 2.

5. Ahh so setting Sx and Sy equal to zero should net me the shape of the rectangular box?

6. Originally Posted by pakman
Ahh so setting Sx and Sy equal to zero should net me the shape of the rectangular box?
Yes! And when you solve for S_x and S_y the solution will probably be in terms of V_o. Meaning, by knowning V_o, which is given, you can find x and y.