Question

1.3(10pts) Find all absolute extrema on the surface f(x,y) = 10y2 – 10x2 subject to the constratint x4 + y4 = 1

Answer 1

salection as Chinen function. ( surface) f(x,y) 104² - 10 oc? ac + + yt. 1 Subjected to constraint let g(x, y) Det + y - 1 Noco. We will Use Lagrange Multiplied Method fcx,y) a Tg(x,y) where, А Lagrange. Multiplier Where fxy) = Gradient of = Gradient- of ffoc,y) <foc, fy> foc a fracy) = ( 1092-1082) ņ θα dx - 10x220 20 x for a da (10 y ²- o se?) 10x2y f = 204

4. v g(x,y) = gradient of gray) 2 3x > Joc ə C30,45) а (х + 4 4) 2 dx OX 4 3 г Jac 35 a (x** - 4) * * * 443 ау үS (1.3) avg(x) <foc, fy? 23x. 8-> 4 - 20 x, 204 ? а < 4x°, 4-5 7 - 20 х , 2017 Кә4 х , Ә 483 » Now. Comparing Components on Both sides . we have following got 3 equations - 20 х а 43 E 2° 5 а 413 а - 20 4х2 а 20 * *

2 92 d A 20 402 20 g2 건 ㅗ 202 y? 2 2 ว + 4 We have constraint 범 키 NaO, Substituting value of y² in constraint tg? 1 2. We have, 야 지? 4 2 at td 1 4 8 2. a 1+ + ( 0S 1부 2. => when X ㅔ일 y S When & y² 위

- Now find value of function A these points , All At 하 and y² : 호 fg) 10x W () 40 고 - 11 AS -OE Point 0c² and y = = 페 네일 10 ( ) - 10() 20 호 0 Thus Absolut Maximum value is 105E X 해 and AS a "고 사 념 t 이 (A)

=) Thus all absolute extrema an Surface Axe C (x,y) (thề, tạº). (CA”, ”) ~ (- (), () And lts, > 헤 (*) 1년