Could someone explain this person's work in more in-depth? I can not follow it. Thank you

4. (10 points) A closed rectangular box has volume 32 can'. What are the lengths of the edges giving the minimum surface area? volume : xyz = 32 / 50 2- 32 8 (xy+ 32 We want to minimize *(xx) = 2(xy+ 32 + 32 subject to *, 430 32, xy 70. (The lost constraint implies * ,$0 critical points : 58x = 24 - 64: 0 32 x2 By = 2 x - 64 32 = 0 Y = 32 Lq 2 X = x'l 32 Jy = ( 32 ) ? X = (32 )13 (where we use ) that *,>0 ) D= fox fry- ty = (128/x2)(128/:) - 47 0 at ( 32#, 32"') and fox ( $245 , 32") 20 - Local minimum of that point. ( It is also a global minimum since f ( xy) + was x my+ 0en Do ) Final anywen : X= y: 2= 3243 cm 3