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5 let f x y z x 2 3 xy 4 y 2 e z 9 a find a point x y

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5. Let f ( x, y, z ) = x 2 + 3 xy + 4 y 2 + e z - 9. a ) Find a point ( x 0 , y 0 , z 0 ) satisfying f ( x 0 , y 0 , z 0 ) = 0 . b ) Can x be expressed as a function g ( y, z ) in some neighborhood of ( x 0 , y 0 , z 0 )? c ) Compute dg ( y 0 , z 0 ).
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MATHEMATICAL ECONOMICS MIDTERM #2, NOVEMBER 12, 2002 Page 3 Answer: a ) The point ( x 0 , y 0 , z 0 ) = (1 , 1 , 0) satisfies the equation. b ) We compute ∂f/∂x = 2 x + 3 y . Plugging in x = 1 and y = 1, we obtain 5. The derivative is invertible (not zero). The Implicit Function Theorem then yields the desired function g . In fact, it is possible to compute g ( y, z ) = [ - 3 y + p 36 - 7 y 2 - 4 e z ] / 2. c ) The derivative is given by dg (1 , 1 , 0) = - 1 5 ∂f ∂y , ∂f ∂z (1 , 1 , 0) = - 1 5 (3 x + 8 y, e z ) (1 , 1 , 0) = - 11 5 , - 1 5 .
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