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# ps1 - by a hyperplane 8 Are the upper contour sets of a...

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Plot the following 2 points on a graph: x = (2 ; 5) and y = (12 ; 11) : 1. Let = 1 = 2 : What is + (1 ) y ? . Graph this new point and call it z: 2. Repeat #1 for the following values of : = : 1 ; : 2 ; : 3 ; : 9 : Call each point z 1 ; z 2 ; z 3 ; and z 9 3. Prove that if a set S is closed, then its complement S c must be open. 4. What is the MRS for the following utility funciton? u ( x; y ) = x 2 + y 2 5. Is the set [2 ; 9) bounded? Prove it. 6. Is the set [2 ; 9) compact? Prove it. 7. What must be true about two sets P and Q if they are to be separated
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Unformatted text preview: by a hyperplane? 8. Are the upper contour sets of a convex funciton convex? Prove it or show via example. 9. Set up a Lagrangian and solve the following constrained maximization problem for ( x & 1 ; x & 2 ) : max u ( x 1 x 2 ) = x 1 p x 2 subject to p 1 x 1 + p 2 x 2 ± 100 10. Do you have any money remaining at the solution to #9? Explain, prove or show. 1...
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