hwsol_17

hwsol_17 - Homework Assignment 17 - (12.7) - Solutions Page...

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Homework Assignment 17 - (12.7) - Solutions Page 991 : 6*, 8*, 10*, 38*, 40*, 46*, 51-54 6. f ! x , y " ! cos 2 x " y 2 ! f ! x , y " ! " 2cos x sin x ,2 y ! " sin ! 2 x " y ! 0 # # sin ! 2 x " ! 0 y ! 0 sin ! 2 x " ! 0if2 x ! n ! , x ! n ! 2 , n ! 0, $ 1, $ 2, . .. Critical points: n ! 2 ,0 , n ! 0, $ 1, $ f xx ! " ! 2 x " , f yy ! 2, f xy ! 0, D ! x , y " ! " 4cos ! 2 x " D n ! 2 ! " ! n ! " ! 4 % 0if n ! 2 k " 1, k ! 0,1,. " 4 & n ! 2 k , k ! When n ! 2 k " 1, f xx ! 2 k " 1 " ! 2 ! " !! 2 k " 1 " ! " ! 2 % 0. So, points ! 2 k " 1 " ! 2 are local minima and 2 k ! 2 ! k ! are saddle points. 8. f ! x , y " ! 4 xy " x 4 " y 4 " 4 ! f ! x , y " ! 4 y " 4 x 3 ,4 x " 4 y 3 ! 0 # # 4 y " 4 x 3 ! 0, y ! x 3 4 x " 4 y 3 ! 0, 4 ! x " x 9 " ! 4 x ! 1 " x 8 " ! 0 x ! 0, x !$ 1 Critical points: 0, 0 , 1, 1 and " 1, " 1 . f xx ! " 12 x 2 , f xy ! 4, f yy ! " 12 y 2 , D ! x , y " ! ! " 12 x 2 "! " 12 y 2 " " 16 ! 144 x 2 y 2 " 16 D ! 0,0 " ! 0 " 16 & 0, so, ! " is a saddle point. D ! 1,1 " ! 144 " 16 % 0, f xx ! " ! " 12 & 0, so ! " is a local maximum. D ! " 1, " 1 " ! 144 " 16 % 0, f xx ! " 1, " 1 " ! " 12 & 0, so ! " 1, " 1 " is also a local maximum. 10. f ! x , y " ! 2 x 2 " y 3 " x 2 y " 3 y ! f ! x , y " ! 4 x " 2 xy ,3 y 2 " x 2 " 3 ! 0 # 4 x " 2 xy ! 2 x ! 2 " y " ! 0, x ! 0or y ! 2 3 y 2 " x 2 " 3 ! 0 # when x ! 0, 3 ! y 2 " 1 " ! 0, y 1 when y ! 2, 3 ! 4 " " x 2 " 3 ! " x 2 " 9 ! 0, x 3 Critical points: 0, 1 , 0, " 1 , 3, 2 and " 3, 2 . f xx ! 4 " 2 y , f xy ! " 2 x , f yy ! 6 y , D ! x , y " ! 12 ! 2 " y " y " 4 x 2 D !
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hwsol_17 - Homework Assignment 17 - (12.7) - Solutions Page...

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