Question

Exercise 1 Find and classify the stationary points of f(x, y) = (x² + y)e8/2.

Answer 1

1) The given function 9/ és $3,3) =(22+y).e =o 9/2 =0 9/2 2 = 0 2.e +y+2 2 x=0 let 25 = 0 & ab doc dy Y2 = 2xe I €"(x2+4+2) = 0 2 =) (a)=0 90 =) X 3013:20 e) te & o) cet 4+2 = 0 y = -2 So, (0, 2) is the only stationary point 9/2 Mon, ²b 2e әх? ay² + (x+3+2) e So, ²b & 8²0 ay² (0,-2) Co,-2) 9/2 & ab 22 e 2 2 활 e 2x² ze And a²6 = 0 axay 2 So, D 226 oy2 ²t axdy ax² & at ax?

se, cat (0, 2) the function has relative minimce and minimum value es $40,-2) = -2 е 2) Let X , yo the dimension of the base (bottom) q the box. The volume V = 8 xyz = 8 =) Z=8 xy be the height of the box cost The total of materials és : CCx,y) = 120 xy + 40xy + 10 (2x2 + 2yz) 1600g + 2011 _+20 y z 16037+ 206016) +2018 160 + 160 J = 160 8 xy 11 160 + 26 find coutical peints. Next to let ac 0 ax =-_1607 - 160 & ac = = co X² => 160 2 - 160 y? = xy=1 2 ay = 1

I You x = 1 x x3 = 1 x = 1 ㅠ ㅠ 7 2 y = 1 & Z = 8 box is: = 1 So, the dimension q wid length & height = 8 Hope this is helpful