Question

Suppose that f(x,y)=-2x3 - 7 xy + 8y2, (-0.5104, 0.2233) is a critical point, 1 xx l (-0.5104.0.2233) = 6.1250, and D(-0.5104, 0.2233) = 49. Which of these statements describes the graph of f at (-0.5104, 0.2233) ? a) of has a relative minimum value at f(-0.5104, 0.2233) = 1.4626. b) of has a relative maximum value at f(-0.5104, 0.2233) = 1.4626. c) of has a saddle point at f(-0.5104, 0.2233) = 1.4626. d) of has a relative maximum value at f(-0.5104, 0.2233) = -0.9308. e) of has a saddle point at f(-0.5104, 0.2233) = -0.9308. f) of has a relative minimum value at f(-0.5104, 0.2233) = -0.9308. g) None of the above. Review Later

Answer 1

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Aus We know that it f(x,y) is a function with continuous second order partial derivatives frex, fyy and fry at a critical point (de, s).

XX Then let D = Exx(2,8) fyy (2,8) – fxy (2) Now if (a) Do and fax (d,s) >o, then t has relative nomum at (eg). (6) Do and fxx (des) <0, then f has realtive maximum at (es). (c) D o then f has saddle point at de (d) D=0 then No conclusion. Now in our question f(X,Y)= - 2x3 + 89 – 7xy - (-0.5loy, 0.2233) is (vitical point Now I = 6:1250 > 3 Now fox | (-015104, 0.2233) and D(-0.5104, 0.2233) = 490

so Do and fxx >0 at (-015104, 0.2233) so by point (a), f has a relative minimum value at f(-0.5104 0.2233). Now value at x= – 0.5104 33 ،0 = زیبا و X f = -2 (-015104)3 + 8 (0.2233) _ 7 (-0.5 104) (0, 2233) by putting x= -0.5104, y = 0, 2233 in 0] --> F= -2(-0113296336486) + 8 (004986289) + 7 (0.11397232) - 0.26592672972 + 0.39890312 of 0.79780624 = 1.46 26 360 8972 ~ 1,4626

so f(-0,5loy, 0.2233) = 1.4626 Toption @ is correct o I f has a relative minimum value. at f(-0.5 104, 0.2233) = 1.4626