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Tutorial 6

Tutorial 6 - 3 y 2 3 4.3 no.6 Use Lagrange multipliers to...

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MATH1211/10-11(1))/Tu6 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 First Semester: Tutorial 6 1. 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 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 . 2. Identify and determine the nature of the critical points of the functions f ( x,y ) = 2 x 3 - 6 xy
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Unformatted text preview: + 3 y 2 . 3. ( § 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.) 4. Suppose ( x 1 ,y 1 ) ,..., ( x n ,y n ) are some data you obtained, and you want to find an equation of the form y = ax 2 + bx + c (where a,b,c are some constants) to predict the values of y from known values of x . (a) Use the method of least squares to construct a function D ( a,b,c ) that gives the sum of the squares of the distances between the observed and predicted y-values of the data. (b) Find a system of equations that will give the values of a,b,c such that the corresponding curve will best fit the data. (You are not required to solve for the values of a,b,c .) 1...
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