Question

Let f(x,y) = x2 - xy + y2 - y. Find the directions u and the values of Duf(1, -1) for which the following is true. a. Duf(1, - 1) is largest c. Duf(1, - 1) = 0 e. Duf(1, -1)= -3 b. Duf(1, -1) is smallest d. Duf(1, -1) = 4 a. Find the direction u and the value of Duf(1, - 1) for which Duf(1 - 1) is largest u=[i+Oj and D,f(1, - 1) = 0 b. Find the direction u and the value of Duf(1, -1) for which Duf(1, -1) is smallest. =i+ Oj and D,f(1,-1) = 0 c. Find the direction(s) u for which Duf(1, -1) = 0. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. u= (Simplify your answer. Use a comma to separate answers as needed. Type your answer in terms of i andj.) O B. There is no solution. d. Find the direction(s) u for which Duf(1, -1) = 4. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. u= (Simplify your answer. Use a comma to separate answers as needed. Type your answer in terms of i andj.) OB. There is no solution.

Answer 1

flyg)= n2_ny +42_4 Nf l Cagl- yg @ +(-4+04-1J j at (1,-1), Of3î -4 § las. Ouf (1,1) is largest in rosmal direction, so u= 3ê - u Du f1,-1) = (31-4; J. (3î -uj) a - = 5 '5 Uoj. Duf (1,-1) is small est in-of (1,1) direction, u=-3î tuja Dufci, t) = (3-43).(-3+uſ). -6 е и Dic. let us rêt gŷ Duft,t) = (3^-45). (nity ſ) notre =0 34-4400 34 = UY to u= u ê t3

(d). let Du H1,-1)=4 for some 4= nety9 (31-49). (hitys) au Tudtye = 4 34-44 = 4 Snarya square both sides an2+ 1642 Juny = 16x2 +10g) 400 = 24/५ 743-249 u= au ľ_7 . (e), let De fl1,4)=-3 uznity ĵ (34_4%). (rityſ) 5-3 . Justy - 31–49 = -30 18+ya? squaring both sides qu211642_ gung= ano +9y2 - 742=auny, 792241 a=7ît au