I have a question on a homework assigment that says the following:
Design a rectangular box of width w, length l, and height h, which will hold 512 cubic centimeters. The sides of the box cost $0.01/square centimeter and the top and bottom cost $0.02/square centimeter. Find the dimensions of the box that minimize the total cost of materials used. Round these values to the nearest hundreth of a cm. Also show the second partials test to verify that you obtained a minimum, and state the actual minimum cost.
So, what I did was determine my minimizing function and my constraint. I have my minimizing function as V=lwh and my constraint as .04l + .04h + 2w = 512. I then find when my first partials are equal to 0 to find critical points. Then on that critical point, I use the second partials test to see if it was a minimum, but it wasn't. When I use the second partials test, I keep ending up with it telling me I found a maximum. I need some help into why I am coming up with a maximum result instead of a minimum result.
Or is my minimizing function wrong? Im wondering maybe I should minimize V = l + w + h which yields the perimeters instead of V=lwh which yields volume? That just occurred to me now...because I don't neccessarily want to minimize volume.
Thanks in advance,