University:
High School
Course:
High School Mathematics
Academic year:

2023
Views:

246

Pages:

6
Author:

Daisei Horiguchi

Lagrange Multipliers Suppose we want to maximize fixy) subject to a constraint guy)=k, KEIR. This happens when a level curve f(x,y)=C, CEIR, just touches the curve. glx,y)=k. Otherwise we can increase c. (see the figure for an example.) So the curves that just touch have a common tangent line at (xo, yo) the point where they touch. So f(x,y) = g(x,yo) for λειR. A similar argument applies to minimizing f(x,y) subject to a constraint g(x,y)=k. Moreover, the argument extends to find extreme values of f(x,y,z) subject to a constraint glxy,z)=k. Method of Lagrange Multipliers (shown for flay,z) but works for flay) also?) To maximize or minimize f(x,y,z) subject to the constraint glx,y,z)=k (assuming the extreme values exist and go on glx,y,z)=k), do the following. 1) Solve the system of eqs. f(x,y,z) = g(x,y,z) for x,y,z, and 2. ( g(x,y,z)=k 2. Plug all solution pts. from #1 into flx,y,z). The largest value is the max value of f. The smallest value is the min valve of f. Note: A is called the Lagrange Multiplier. ex. Find the extreme values of the function subject to the constraint(s). a) f(x, y) = x ^ 2 - y ^ 2 x ^ 2 + y ^ 2 = 1 Rightarrow g(x, y) = x ^ 2 + y ^ 2 Vf=1 =<2x,2y> > <2x-2y>=<2x,2y> 2x = lambda ^ (2x) nabla f <= 2x_{j} 29> - 2y = lambda * 2y System of eps. - 2y = lambda ^ 2 * y x ^ 2 + y ^ 2 = 1 Nik: Xey can't happen because of 3 1. Solving the system of eqs: Case 1: x=0, y≠0 Care 2: X≠0, y=0 Case 3: X70, y≠0 (3 Rightarrow (0) ^ 2 + y ^ 2 = 1 →(0,1) y = plus/minus 1 Rightarrow y ^ 2 = 1 ( 3 -> x ^ 2 + (0) ^ 2 = 1 Rightarrow x ^ 2 = 1 ⇒(±1,0) Rightarrow X > plus/minus 1 01 lambda = 1 be I AND-1. no sol. 2 -> lambda = - 1 2. Plug (0,±1) and (11,0) into f=x²-y²: f(0, plus/minus 1) = (0) ^ 2 - (plus/minus 1) ^ 2 = - 1 f(11,0)=(±1)²-(0)2 = 1 → max value = 1 Min value=-1 b) f(x,y,z)= exye; 2x ^ 2 + y ^ 2 + z ^ 2 = 24 Rightarrow g(x, y, z) = 2x ^ 2 + y ^ 2 + z ^ 2 2 BAY = JA = =>< > < 4x, 24, 2z> →System of eps: yze=14x0 xze kyz xye ty² = 12z 3 2x ^ 2 + y ^ 2 + z ^ 2 = 24 Note 4 implies x - y = z = 0 Can't happen. ,, and imply just one of xy, andz is o can't happen. 1. Solve the system: Case 1: two of x, y, and 2 areo: x Rightarrow plus/minus 2 * sqrt(3) Rightarrow x ^ 2 = 12 x702x² = 24 >(9 Rightarrow y ^ 2 = 24 > y = plus/minus 2 * sqrt(6) z ne0 (9 Rightarrow z ^ 2 = 24 Rightarrow z = plus/minus 2 * sqrt(6) ⇒ (plus/minus 2 * sqrt(5), 0, 0), (0, plus/minus 2 * sqrt(6), 0) (0, 0, plus/minus 2 * sqrt(6)) Cave 2: x, y, z≠0 → yzexyz xze Kyz xye kyz = 22 → 4x^ 2 =2y^ 2 Rightarrow 2z ^ 2 = 2y ^ 2 Theta(S)G Rightarrow (y) ^ 2 + y ^ 2 + (y) ^ 2 = 24 =±2√2 beta -> lambda = (xy * e ^ (x * y ^ 2))/(2z) 2x^ 2 =y^ 2 ( y -> z ^ 2 = y ^ 2 * (6) y ^ 2 = 8 3y ^ 2 = 2y =ty Rightarrow x = plus/minus y/(sqrt(2)) == =±2 = plus/minus (2sqrt(2))/(sqrt(2)) ⇒(±2,12√2,±2√2) 2. Plug the sol. pts. into f(x,y,z)=exyz. f(12,±2√2, ±2√2) = e±16 f(plus/minus 2 * sqrt(s), 0, o) = f(o, plus/minus 2 * sqrt(6), o) = f(0, 0, plus/minus 2 * sqrt(6)) = 1 max value: elb Min value: e-16 c) f(x,y,z) = x+y+z; x ^ 2 + z ^ 2 = 2 x + y = 1 Two constraints! Use If=1g+Mh. →gla,y,z) = x²+z², h(x,y,z) = x+y ⇒ system of eps.: 1 = lambda * 2x + mu ><| i | i |>= lambda < 2kappa ,0,2z>+ mu<| i | j0 > W=1 I = lambda * 2z x ^ 2 + z ^ 2 = 2(y) x + y = 1 1. Solve the system: (2) Rightarrow mu = 1 +10=72XX=0 (since ③ implies 1≠0) (9) ^ 2 + z ^ 2 =2 Rightarrow z^ 2 =2 Rightarrow z= pm sqrt z (S) (o)+y Rightarrow1 Rightarrow y = 1 → (0, 1, plus/minus sqrt(2)) 2. Plug the sol. pts into f=x+y+z: f(0, 1, sqrt(2)) = 0 + 1 + sqrt(2) = 1 + sqrt(2) f(0,|- sqrt 2 )= 0 + i - sqrt(2) = i - sqrt(2) max value: IH2 min valve: 1-52 > x=-2, y=2 (-22) [sme (2)²+(2)²=8<9, (-2,2) is in the region x ^ 2 + y ^ 2 < 9 ] d) f(x, y) = x ^ 2 + y ^ 2 + 4x - 4y ; x ^ 2 + y ^ 2 <= 9 First extreme of f on x ^ 2 + y ^ 2 < 9 fx=2x+4 fy=2y-4 0= 2x+4=0 →→ f(- 2, 2) = (- 2) ^ 2 + (2) ^ 2 + y(- 2) - 4(z) = - 8 Then find the extreme values of f on x ^ 2 + y ^ 2 = 1 <2x+4, 2y - 4 >= lambda < 2x, 2y > ⇒ system of eys: 2x + y = lambda * 2x 2y - 4 = 22y x ^ 2 + y ^ 2 = 1 O Rightarrow lambda = (2x + 4)/(2x) = (x + 2)/x Q -> x ^ 2 + (- x) ^ 2 = 9 -> 2x ^ 2 = 9 Rightarrow (3/(sqrt(2)), - 3/(sqrt(2))) ,(- 3 sqrt 2 , 3 sqrt 2 ) ⇒ y=-x= F f( 3 sqrt 2 1 - 3 sqrt 2 )= (3/(sqrt(2))) ^ 2 + (- 3/(sqrt(2))) ^ 2 + 4(3/(sqrt(2))) - 4(- 5/(sqrt(2))) = 9/2 + 9/2 + 12/(sqrt(2)) + 12/(sqrt(2)) = 9 + 24/(sqrt(2)) +(-)=()²+()²+4(-)-4(금) = 올+올- -=9--7.97 Theta Rightarrow lambda = (2y - 4)/(2y) = (y - 2)/y x = plus/minus 3/(sqrt(2)) Rightarrow (x + 2)/x / ((y - 2)/y) -> xy + 2y / x * y - 2x 2y leftarrow-2x y=-x Note: imply x = 0 or y = 0 Can't happen. inplies x=y=0 can't happen. max valne: 9 + 24/(sqrt(2)) min value: -8