# There is no global maximum in fact lim x f x 0 there

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4) is a local minimum. There is no global maximum. In fact lim x + f ( x, 0) = + . There is also no global minimum since lim x →-∞ f ( x, 0) = -∞ 3. Consider the quadratic form Q ( x, y ) = x 2 + 8 xy + y 2 . a ) Does the quadratic form have a maximum or minimum? Answer: This quadratic form has associated symmetric matrix A = 1 4 4 1 .

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MATHEMATICAL ECONOMICS MIDTERM #2, NOVEMBER 8, 2001 Page 2 The first leading principal minor is 1 > 0, while the second leading principal minor is 1 - 16 = - 15 < 0. This violates the sign patterns for positive or negative definite, so the matrix is indefinite. The form has neither maximum nor minimum. b ) Now impose the constraint 2 x +3 y = 0. Does the quadratic form have a constrained maximum or minimum? Answer: In this case we consider the bordered matrix: H = 0 2 3 2 1 4 3 4 1 . The only princial minor to consider is the determinant of the whole matrix, which is 35 > 0. Since n = 2 and so ( - 1) n = 1, we have ( - 1) n det H = 1. This implies the form attains a maximum at (0 , 0). 4. Suppose that utility is a continuous function from R 2 + to R . The budget set is { ( c, l ) : ( c, l ) 0 , c 1000 - 10 l, c 100 - l/ 9 } .
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