I'm having problems with optimization, the book calls this section optimization II.
Packaging . By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. if the cardboard is 15 in. long by 8 in. wide, find the dimensions of the box that will yield maximun volume.
I basically drew a box and cut away its 4 corners and labeled each cutaway as "x", and labeled the sides as 15-2x and 8-2x, since we are looking for volume V=lwh
V=lwh= (15-2x)(8-2x)(x) = 4x^3 - 46x^2 + 120x
f ' x= 12x^2 - 92x + 120
Ok, so this is where I get stuck, the example problem similiar to this one has one to look for an interval [ ] to test on. ex. [0, 5]. They mention something about the box length has to be positive, which makes sense, but I don't understand how I get the interval and how I plug it in.
The interval, if I looked at the example problem correctly, comes from this...
15-2x equal or greater than 0
8-2x equal or greater than 0
x equal or greater than 0
Thank you for anyone who can help.