Question

Consider the function f(x,y) = xy - 3x-2y2 + 17x + y + 37 and the constraint glx.v) = -6x + 3y - 12. Find the optimal point of f(x,y) subject to the constraint g(x.). Enter the values of, y. f(x,y), and below. NOTE: Enter correct to 2 decimal places y f(x,y) A-

Answer 1

Here f(x,y) = xy-3x - 2y² + 17x + y + 37 Constraint glx,y) = -6x + 3y = 12 = guy) - 62 + 3y -12 We need to apply Lagrange's Multiplicis to find optimal solution such that

ab txry) = Аў8 Dry) ЧЕ () - (Бя, 4) Бx = у – 62 +1 Бу: 2 – Ҷу +1 9. – у = Substituting these values in Ф Үч-бап), чун 1) - A(-6,3) - 6 2

compoully coefficient and d=2-44 tl y - 6x + 17 =-62 2-4y +1 = 31 > x=y-6x +17 -6 Comparing both > y-62 +17 - x-4y+ 3 - -6-2 y-6x+17= 2 CX-49/1) 4-62 +17--23 +8y-2 y-bat 17+2x-84+2=0 -42-7y +19=0 u atly-1950 19 How ga, y) = -6x + 3y =12 -62 + 3y-12-0 Dividing by 3, axty-40 by a we get - 4x + ay - 8 = 0 6 os and multiplying

18 - 11 4 40 у 2 2 2 Ч flacy) = -3 bc.y) = -3 1 3 4 22 toy) -20 22 3 Дcx,y) = -10422 #3 12 -3 Ч 9 b tx,y) = 48 -3 - Чs 1. 8s Ч 2 ч Ч Summarizing we get ю.5 or 2. y=3 b(x,y) = 45 БҮ 11. as Х: - 3.83