Question

61. (14.8) Find the extreme values for the function 8y - 4:subject to the constraints y2 + x2 - 1 = 0 and 2x - z-y-2 = 0.

Answer 1

let f(x,y)= 84 - 4x 9(8,9,2) = 27x²1=0 h (2,9,2) = 2x-2-4-2=0 all want to ind extreme value we of f(x, y, 2) of thash Subject % ст.9,) с4 htt, z) So first finding all critical points on constraints fx=-4, ty=8 , fyty fz=0 gx = 2x gy= 2y = 0 hx=2 hy=nt, he = -1 using Lagrange Multiplier fx = dgn tuhr fy= agytuhy fe=aga teha g(419, 2) = 4²4x²,20 hlx,y,2)=2x-2-4-2=0 &, м а у –4=dx2x+212, 8-a gytu (1), 0- 2xotul-1) - l=0 l=0 » ~4=22x and 8=a24 » F2=ax and (4=ay _ y?tx-1=0 and 2x-2-4-2=0 ti Clearly a to, xa 2 ant y = 4 Putting this in equahm o Since - 90=a² .=) a = ± 520-1255 and putting x=3 and y=4 in egaarm ③ we get - -4-2-4-2=0 - Z= -2 -2 So, we get 1=2 7=4 122-54-2

d 외 85 | TE A = 2.55, hon x- 13-2 2 .) - - , Z : -409 | 50( 45) 1. 15. k u- -2 J5 ) ( 55- -4플용 여1훓1볼 (무) -() 48 = CG, 플무) Is - - 음 =-4 JS_- - so, Extreme value are - Iman = 455 fmin=-455 and