The math doesn't lie: if there is no maximum, then it is theoretically possible to use an infinite amount of material to build a box of that volume, unless there are any bounding values for the individual dimensions.
Here's another question Ive come across during revision
So
Volume is
Total area is base of box + 4 sides -
And then I got the derivative of
And where this derivative = 0, there was a local minimum. This was where w = 40, so that makes d = 40 and h = 20 for the dimensions of the box that will use the minimum amount of material.
I take it this is all correct?
If it is, what I'm wondering is how I would be able to find the dimensions that would use the maximum amount of material? I only got one critical point when differentiating and that turned out to be a local minmum so how is it possible to find the maximum amount of material used?
For example, if you were to make the length of a side of the base, w, equal to 0.0001, then the height would be and so the total surface area would be 0.000001+ 4(32000000000)(0.001)= 128000000.000001. By making the base smaller and smaller, you can make the height, and so the surface area, arbitrarily large.
More generally, the derivative for f(w) you got was . Yes, f'(40)= 80- 80= 0. If w is larger than 50 then w- 40>0 and the other numbers are clearly positive so f'> 0. That is, f is increasing for all w> 40 and so there is no maximum surface area.