find the dimensions of the rectangular box of least surface area that has a volume of 1000 cubic inches. (@ denotes partial derivative)

1. V = lxy = 1000 .. l = 1000/xy
2. f(x) = 2ly + 2lx + 2xy = 2(1000/xy)y + 2(1000/xy)x + 2xy = 2000/x + 2000/y + 2xy
3. @f/@x = -2000/x^2 + 2y
4. @f/@y = -2000/y^2 + 2x
5. -2000/x^2 + 2y = 0 .. -2y = -2000/x^2 .. y = 1000/x^2
6. -2000/y^2 + 2x = 0 .. -2000/(1000/x^2) + 2x = 0 .. 2x^2 = 2x .. x = 1
7. y = 1000/x^2 = 1000/1^2 .. y = 1000
8. l = 1000/xy = 1000/1*1000 .. l = 1