Economics 204–Final Exam–August 21, 2007, 9am-12pmEach of the four questions is worth 25% of the totalPlease use threeseparatebluebooks, one for each of the three PartsPart I1. State and prove the Supremum Property.2. Prove that if a finite setXhas exactlynelements, then 2X, the set of all subsets ofX, has exactly 2nelements.Hint: use induction.Part II3. Consider the functionf(x, y) =x3−y3+ 2x2+ 2xy+ 8y2−9x−15y−2(a) Compute the first order conditions for a local maximum or minimum off. Showthat the first order conditions are satisfied at the point (x0, y0) = (1,1).(b) ComputeD2f(x0, y0) and give the quadratic Taylor polynomial forfat the point(x0, y0).(c) Find the eigenvalues ofD2f(x0, y0) and determine whetherfhas a local max, alocal min, or a saddle at (x0, y0).(d) Doesfhave a global max, a global min, or neither, at (x0, y0)?(e) Find the eigenvectors ofD2f(x0, y0) and provide an orthonormal basis forR2consisting of eigenvectors. Rewrite the quadratic Taylor polynomial forfat thepoint (x0, y0) in terms of this basis.
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