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Unformatted text preview: as possible. Which point or points on the island qualify? 5. [10 marks] Classify the critical points of the function f : R 2 R , ( x, y ) m ( x 2)( x 2 y 2 ) . 1 6. [10 marks] Suppose that U is the nonconvex set U = { ( x, y ) R 2 : y < x 2 + 1 } and f : U R is a C 1 function such that b f ( x ) b 3 for x U . Using the Mean Value Theorem, show that f (3 , 4) f ( 3 , 0) + 30. 7. [10 marks] DeFne g : R 2 R , ( x, y ) m sin( x 2 ) cos( y ) . ind g (0 , 0) for all multiindices of order six. Note: Once you have a numerical answer, you need not simplify it. That is, 4! / 2! is just as acceptable an answer as 12. 8. Bonus Question: [5 marks] Let A be an n n positive deFnite matrix and B j an n n positive semideFnite matrix for j = 1 , . . . , . Show that the sum A + s j =0 B j is a positive deFnite matrix. 2...
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 Fall '09
 RomauldStanczak
 Calculus, Derivative, Multivariable Calculus

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