s7 - Solution to Set 7 1(a fx = 2x 6 fy = 4 y 8 fx = fy = 0...

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Solution to Set 7 1. (a) f x =2 x 6 f y =4 y +8 f x = f y =0 x =3 ,y = 2T h e c i r i t c a l p o i n t i s ( 3 , 2). f xx f yy f xy D = f xx f f 2 xy =8 > 0& f xx > 0 The point is a relative minimum. (b) g x = ye xy g y = xe xy g x = g y x = 0 The ciritcal point is (0 , 0). g xx = y 2 e xy g = y 2 e xy g xy = e xy + xye xy At (0 , 0) ,D = g xx g g 2 xy = 1 The point is a saddle point. (c) f u u 2 6 vf v v 2 6 u f u = f v v = u 2 2 & u 4 4 2 u u ( u 3 8) = 0 u =0or2 The ciritcal points are (0 , 0) and (2 , 2). f uu =6 uf vv uv = 6 D = f uu f vv f 2 uv = 36( uv 1) At (0 , 0) = 36 < 0 The point is a saddle point. At (2 , 2) = 108 > f uu =12 > 0 The point is a relative minimum. (d) f x = x p 1 x 2 2 y 2 f y = f x = 2 y p 1 x 2 2 y 2 f x = f y x = 0 The ciritcal point is (0 , 0). Note that f x ,f y are undefned at the boundary oF the domain oF f , but the boundary points need not be counted as critical points. f xx = 2 y 2 1 (1 x 2 2 y 2 ) 3 / 2 f = 2 x 2 2 (1 x 2 2 y 2 ) 3 / 2 f xy = 2 xy (1 x 2
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s7 - Solution to Set 7 1(a fx = 2x 6 fy = 4 y 8 fx = fy = 0...

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