Question

Find the absolute extrema of f(x, y) = x^2 + y^2 − 2x − 2y + 1 on the set D = {(x, y): 0 ≤ x ≤ 2 , 0 ≤ y ≤ 2 }

Answer 1

fy)= 2² + y ² - 2x - 2y + LD differunale Lille ? ? 10202 lingy Of content f 9X022.1. 0:20 Differentiate again with & treating y As constant fx= 2x1-0 for F2 = 8 - referenhase so o wie z preaking in a constant fyr O+ 24-0- 2x1 + b G & trating . Again differentiate wry yn (on fton fyys_ax: 0 to formy - ©

Differentiate Eq? 27 wrky & koeping. a les constant fry= 240.0 fry : b. So

from &Q2 (2 plut ft: 0 22-2=0 HEL from Exo , put. 2 - 250 Y=L We have paint (x, y) = (1,1) These valves dies in the set of (2,7) values (ny - OLXL2, OLYLL given paint So we have costica for 22 Co 4 2 . for Y = (0, 1, 2

Istoking for et horum Yelise. If at-520 The then flf17) has a mat vore 870 then f(x, y) hay a minm vous So here in this problem 870 ie 270 & ots² = 2x20 at-s2 = 470 So this lunchon har alloy TamUM Value: Now all have different set of Chhcel point Txing 60,0), (011, (0,21811,02, 113,02) 1914 19.2)

All have funchon f(x1y)) x2 + y ² - 2x - 28 + 1 Cet (0,0) f(0,0) = 1 at (0 ) flo, 1) = 0+1-O-2 tl. at (0, 2) f(0, 2) lo.of u-o-2x2t) 5.) 4-42 at (10) fllo). 170.-2-ot! at 0.0 f) = -2-2 :) - at (112) (,), 144- 2- 41 Cet (2,01 of 2107 440-4.0+1

at (211) 4+1-4-2 + f/2, 1) = et (212). f (212) = 4+4-4-4+1 EL por from cufferent. Wobre of fluny) uce see that abr Minm valire a f(x, y) = -1 IDCC Noter according to ricle there is no abr. Marry valcre, Gut from given value of 24,9) рье. Махму) = 1