Question

PROSEC Find all relative extreme points and saddle points of f(x, y) = x - y2 - 2xy.

Answer 1

. Given that flny) = y ² any Here First of all we have to find all first and and order partial derivative Fu= 3 ay B and fun on - fy= -24 - on and fyy = -2 and also Fny = -2 critical points Now for put Fuzo and fy=0 Fuzo ; 34² gy=0 24 = 34² 6 fy=0 -gy-qu=0 2y = -que - row put value of gy from ea o in to ea 34² = - 2x - 2x sultan so a( 34+2) =0 n=0 OR 34+2=0 [n=0) OR (r= -2

For value of y then When n o y=0 and when u=-2 then ya g it rom eq @ 29=- 24 39 u =-y . y=-u - so critical points are (0,0) and (-2/2 2 2 ) at point (0,0) Fase 6x0 = 0 Fyy = -2 -2 Fay = Now using D-Test D1010) - Fux (0,0). Fyy (010) - (Fuy (0,0))? conditions o if D(0,0) >0 and Fun 80,0) 70 then there is a local minima at point (0,0) (Relative) if Dio, to and free (010) co then there is a does ( кеlаtіе) мамі мя at po54 (oa) . if D (0,03 Lo then point 1010) is saddle o it or 0,0) =0 then there is no minima / maxim rs Scanned with

Now let we checkin D(0,0) = 0x-2-(-2) ? = cuco so point 1010) is saddle point nrow at point la 01-31, ) = Fuel , ). Fay ( *) - (Fay (-2, 0); poi fun= fue 9 Fjxxel *** })= 6*= -4 Fyy { }, })= -2 Fay 1 - 2 / 2 / 3 = -2 NOL D! )= -48-2-(-e) ol )= y so and fun a-4 Lo $0 point = 2) is point of manima.