Maxima and minima section

• Jan 29th 2007, 04:39 PM
pakman
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.
• Jan 29th 2007, 04:49 PM
ThePerfectHacker
Quote:

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,
$\displaystyle S_x=0$
$\displaystyle S_y=0$
• Jan 29th 2007, 05:06 PM
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.
• Jan 29th 2007, 05:09 PM
ThePerfectHacker
Quote:

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.
• Jan 29th 2007, 05:14 PM
pakman
Ahh so setting Sx and Sy equal to zero should net me the shape of the rectangular box?
• Jan 29th 2007, 05:17 PM
ThePerfectHacker
Quote:

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.