tutorial5

tutorial5 - counter-example.) (a) If f has a local maximum...

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MATH1211/10-11(2)/Tu5/TNK THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 2nd Semester: Tutorial 5 Date of tutorial classes: March 10–11. (The section/problem numbers in the following refer to those in the textbook.) 1. ( § 4.1 no.4) Find the Taylor polynomial p 3 of order 3 of the function f ( x ) = x at the point a = 1. 2. Let f ( x,y,z ) = x 2 cos y + y 2 z . (a) Calculate the derivative matrix and the Hessian matrix for f at the point (1 , π 2 , 1). (b) Express the second-order Taylor polynomial of f at (1 , π 2 , 1) by using the derivative matrix and the Hessian matrix as in formula (10) on p.241. 3. Let f ( x,y ) , g ( x,y ) : R 2 R be scalar-valued multivariable functions. Determine whether the following statements are true or false. ( Remark : To show that a statement is true, you must give a proof for the most general case. To show that a statement is false, the best way is to give a
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Unformatted text preview: counter-example.) (a) If f has a local maximum at the point a , then f cannot have a local minimum at a . (b) If a is a saddle point of f , then f cannot have a local extremum at a . (To see the definition of a saddle point, please read Example 1 on p.246 and the description in lines 1–2 on p.247 of textbook.) (c) If f has a local minimum at a and g has a local maximum at a , then a is a saddle point of f + g . 4. Identify and determine the nature of the critical points of the functions f ( x,y ) = 2 x 3-6 xy + 3 y 2 . 5. ( § 4.3 no.6) Use Lagrange multipliers to identify the critical points of f ( x,y,z ) = x 2 + y 2 + z 2 subject to the constraint x + y-z = 1. (Note: Here you are not required to characterize the critical points you found.) 1...
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