Question

9) Find the absolute maxima and minima of the function f(x,y) = x2 + xy + y2 on the square -8 < x,y 5 8

Answer 1

Consider given function f(x,y) = 2? +ory+y2 which is continuous on set D = {cx, 431-8<x<8, -84438} Diagramatically, D is described as, ny 8 8 x -8 Now this Square has following conditions, upper side ; y=8, -84948 lower side: Y--8, -8 <258 left side ; X=-8, -8%y$ 8 right side : =8, -85Y 68 Also it is closed We find critical points inside square so take partial derivative of ean with respect to x, of fx = (x²+xy tye) = 22ty y as 2x af fy Oy (Q2Fcy +99 Ea+2y dy CO For critical point let fx co, fyz 2x+y=0 28=-4 x + 2y co e +26-28)=0 (+2282-y) as 0-48 ao a-3x20 atazo Усо is only critical point inside square (x,y) = (0,oo

6 Further f(0,0) = o toto?= 0 f(0,0)=0) Next we find critical points on boundties – For upper side: Y = 8, 85258 Define new function, 9,(x) = f(2,8) x²+88+64 To find critical points Ta pautahad derivative of g, (a) with respeck to x 2288 9,60) 24 (9,ca))= a (x2+82 +84) - conside gỗ Cx) co 2xt8 CO -4 is critical point value of g(6) at this critical point and at end point as, x=-4*9(-4) I= -8, +968) 5 64 9(8) a 192 = 48 XE je 96-4) = f(-4,8) = 48 96-8)=f(-8,8) 264 109) gCl) = F(8,8) 5192 2 To lower side: y=-8, +2CXL8 Define qq (x) = f(x, -8) = x² – 8x + 84 find critical point Take sand derivative of 92 (0) with respect to a gcXc -(22-88+64) E 22-8 2x Treate 9.(X)=0 2X-8 EU > 1*34] is crasitical point

Value of 9G) at this critical point & and at endpoints as k=4, 9264) = 48, a=-8 » 9268)_192 H28, 92(8) = 64 ,, 9264) - F14,-8) - 48 92C-8) = f(-87-8) = 192 4 92(8) = f(-8,-) - 64 & For Define new function, For criteal point left side: 23-8. -82748 hi(y)=f(-8,4)= y2-87 +64 to y . 24-8 Take parbled derivalt've of hi(4) with respect hicas hicys=0 э] = 4 is critical point Finding value of bicy) ak critical point pak end point YE4 a bic4)=48 ya-8,) hits)< 192 2y & Co Y=8 - h (8) = 64 hich) = f(-8,4)= 48 hic-8) =f(-8,-8)= 192 hi (8) = f(~8,8) = 64 For right side a x=8, -&Lys 8 Define bacy ) F(8,y) = y2 +84 +64 For critical point, Differentiate holy) worth respeck to y hecy) = 2y+8 hecylco 24+8=0 > y=-4 is cortical point

Finding value of ha Cy) at critical point o at end points Do yan4 s hz (64) = 48 & h264) = 48 Y auf & he (-8)= 64 y=8 g ha 8) = 192 ha (-4)= F(8,-4) = 48 ha(-8) + F(8,-8) 264 ho (8) - FC8, 8) 2192 .: From ean 2, 39 ♡ & 5, we get f(0,0)= 0, F(-4,8) = 48 F(8,8) = 64 F(8,8) = 192 f(4,8)= 48 f(-8,-8) = 192 F(8,8) = 64 f(-8,4)= 48 F08,-4) = 48 UM function has maximur value at (8,8) and at (-8,-8) minimum value at (0,0) Absolute maximum of frx,y) 192 and occur at (8,8),(-8,8) Absolute minimum of flaggoro and occur at coo)