1. The problem statement, all variables and given/known data I have 12m2 of cardboard. I must make a rectangular box with no lid, but I must maximize the potential volume of the box. The length, width and height of the box are x,y,z. 2. Relevant equations I know that xyz = V (volume). I know the surface area is 12m2 ∴ 2xz+2yz+xy=12 ∴z=(12-xy)/(2x+2y) ∴V = xy(12-xy)/(2x+2y) I've done the calculations to find the first and second derivatives: ∂V/∂x = -y2(x2+2xy-12)/2(x+y)2 ∂V/∂y = -x2(y2+2xy-12)/2(x+y)2 3. The attempt at a solution I know I must make these = 0 in order to find the critical points. I must then use the Hessian matrix to apply the second derivative test. For ∂V/∂x=0 I got the identity x=(12-y2)/2y I then subbed this into x2+2xy-12 The I got this equation: 3y4+24y2-24=0 I think the y values of this equations represent the y-values of the critical points, but I don't know how to evaluate them. Or maybe I made a mistake somewhere along the way. Can anyone help me with this? I'd really appreciate it.