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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 ∂ α g (0 , 0) for all multiindices α 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 , . . . , ℓ . Show that the sum A + ℓ s j =0 B j is a positive deFnite matrix. 2...
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This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto.
 Fall '09
 RomauldStanczak
 Calculus, Derivative, Multivariable Calculus

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