Question

2. For each function, find all critical points and use the Hessian to determine whether they are local maxima, minima, or saddle points. (a) f(x,y,z) = x — 2 sin x – 3yz (b) g(x, y, z) = cosh x + 4yz – 2y2 – 24 (c) u(x, y, z) = (x – z)4 – x2 + y2 + 6x2 – 22

Answer 1

27 So - 3 Zz 0 ) Zzo 3Y 2 2. a> f(x,y,z) 2 x - 2 Sinx - 3yZ For critical points, af, af, at zo of 1 - 2 cosx = 0 =) ces XzK critical points af are (1/2,0,0), af (-1/3,0,0) 2 z 39 20 =) 720 The Hessian Matrix is a square malsix of second ondered partial derivatives of a scales function. Hessian matrix of 3 variables (xix, 2) af H - af af af 22f auf 24 azax azay der O 14 f(x,y,z)) - -3 - 2 sinx 2sinx (0-9) 2-18 Sinx 27 zef axr axay axaz ayax ayr ayaz Sinx 0 o 0 | f(x,y,z7|(16,0,0) 2 -18 Sin T, 2 -18 x · (W/,0,0) is 2-93co a local maxima 1 $(Xixz> (-17,00) 2 "(-11/3,0,0) -18 sin (-9%) 18x 2 , 95370 :- (-1/10,0) is a local minima 2

X b) g(x,y, 2) a coshx + 4y2-24²-29 +447 - 24 – ZA 29 sinha 2 2 32x20 2x 2x22 29 47 - 4y = = 0z) Zzay afy 244y -47% -0 => Y =23 :. Z 223 Z(E 1) 20 2) Z20, 1 critical points are 10,0,0), (0,-1,-1), (0,1,1) af af af H2 2x axay 22% af 22f ayax ay ayaz auf 227 2 zu cashx O | Hg 1 27/7)| - o -4 o 4 casha 16 22 2x azdy 4 -48372 O o 0 1 -372 = 16 cashx ( 37 -1) | 4 g(x,x23600,0) 119,0,0) = 16 cash o (3x0²-1) = 0 (0,0,0) is a saddle point. | H g(x,7,2)) co,-1,-1) = 16 cash o (3xC-117-1) 20 .. (0,-1,-1) is a saddle poinh Similarly (ortal) is also a sadtle point.

c) u(x, y, z) 2 (x-7)9 _x²+7²+ 6x2 – z2 2x = 4(x-7)3 - 2x +67-0-02 2 2yzo 2z 4Z - 22 27 2) yao 2-4 (x-2)3 + 6x = 0 ;) (i) + (ii) 4x + 20 2) X2-Z from (i) 4 (x+x) 3 - 2x - 6x 20 2) 4 (2x)3 – 8x=0 2) 4.8x3 - 8x20 2) 4x² - X=0 2) X (4x²-1)=0 => X20, X2 +12 So critical points are (0,0,0), 101/60, -1/2), (-/10, 1/2) 24€ 224 af H = axay a.az 2x2 22f әу әх ayr 22f azay az ax 21 22 22 az 0 2 4X3 (x-2) 2 - 2 39x3 (x - 2)² + 6 Laura L-4x3x (x-2)² + 6 443 (x-2)²_2 1 HUCK 1,73|60,0) 13:1-4-361-3760 IH U (X,Y,Z) (0,0,0) is local maxima 12-2 - 12 + 6 1 1 1 10 - 1/) -12 +6 12-2 2,0,-/ 2100-36.6470 (6,010) is local minima Similarly (-2,0 %) is a local minima. 10 2