ps6 - Atsushi Inoue ECG 765: Mathematical Methods for...

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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Problem Set 6 Due in class on Tuesday, October 13 Fall 2009 1. Let A= a11 a21 . . . a12 a22 am1 am2 a1n a2n . . . . . . amn , x= b1 b2 . . . bn . Write Ax using aij , bj and the notation. (1 point) 2. Let f : X where X is an open set in n . We say that f is homogeneous of degree p over X if f (x) = p f (x) for every and for every x X for which x X. If such a function is differentiable at x, show that < x, f (x) >= pf (x). (2 points) 3. Consider the following function f (x, y) = xy x2 +y 2 0 if (x, y) = (0, 0) if (x, y) = (0, 0) Compute the partial derivatives fx and fy when (x, y) = (0, 0). (2 points) 4. Let f : + + be a C 1 function that satisfies f (0) = 0 and limx f (x) = 0. Suppose that there is only a single point x + at which f (x) = 0. Show that x must be a global maximum of f on + . (2 points) 5. Is (x, y) = (0, 0) a local maximum, local minimum or inflection point of each of the following functions? (a) f (x, y) = x2 + 4xy + y 2 . (1 point) (b) f (x, y) = x4 + y 4 . (1 point) (c) f (x, y) = x4 - y 4 . (1 point) 1 ...
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This note was uploaded on 10/22/2009 for the course ECG 765 taught by Professor Fackler during the Fall '07 term at N.C. State.

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