Question

1. Find the extreme values of the following functions.(Write together the problem-solving process.)

(1) f(x, y) = x^2 + 2xy + 2y^2

(2) f(x, y) = x^2 + 2xy − 2y^2

(3) f(x, y) = 3x^2 + 6xy − 2y^3

Answer 1

Given, f(x, y) = x + 2xy +242 (0) fx = 2x+2y by = 2x+44 Exy = 2 for fyy To find critical points and ato = 3 +2=0 x4y= zo 2xthy 2(-4) + hy so zy=0 -fyzo X-5 =>x=0 (60) is a critical point. 2 = (2) (14) . (2) 2 2 fxx. fyy 11 2 and fyy at 10o) : fxx.fy - key fror. fyy - Fry 8-4 fax-tyy-frey • fyy-frey of fxx =2 forex>0} shtyy bas from Oband ☺ f (24) minimum value at cool. if(x4) = x²+2xy + 2y. f(0) = (0)2+2(0) (0) + 26032 - 110,0)=0 . The minimum value of f(x, y) 6 the

(2) f(x4y)= ge²+ 2xy-se? - fx=2xt ry fy = 2x-44 fxy = 2 2 fxx=2 ztrix out fyy = -4 To find critical points :- fu and fy ze = antzyzo and 2x-44 xzuy 21-4) -lyco { x = 4 -by-ou-( yzo as neg x = -(0) [yzo] lo the critical point 2 (2) (-4) (2) at (oo) : foon. Tuy fry fxx.fyy-fry fux-fyy-towy 8-4 2 = fx. fyy-frey saddle point & Cool is function f(x, y) of the i No extreme values

3 f (x,y) = 3x²+ory-243 fx = 6x + by 6x-by² fy fuy = 6 HSL fox = 6 fyy = -124 To find critical points fx=0 and fyzo 6x + 6yo xty to 2=-4 2-0 6x- by LE where fyzotning shoe 21 bu-by ² = 0 [:x=-9] (-) a to → 6ry) - 6yzzo ģ-by-by²zo - 6y (1+y) = y=o and ity=0 y=- i as x=- at x = -10) = Nzo tesort be at y=-1: N =-(-1) => x = 1 B(1,1) 1 the and Bus) ane A (0,0) critical points .

at A (oo) - fyy for = 6 = -124 =-1210) = 0 fyy fny اع fox foly frey (0) (0) -(6)? = 0-36 =-36 (fx. fyy - bez A(0,o) is p sadde point at B(1,-1): frox = 6 fyy 2-12y frey = 6 in frox. fgy - taj 2 = (6) (12) - (6) = 72-36 fyy E12 - 12 (1) = 12 ) 2 = 36 2 = 6 سقم fyy = 12 frost fyy-fry > o fon faxx >0 and fyys of from ① and 2 and , f(x,y) has a. minimum at B ( 17) f(0, -1) = 343 764) (-)-2(-1}} E f(xY) = 33+ +6 y 34 zey? = f (1,-1) =-) minimum value of f(x,y) = 3-6+2 is -1