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LSweek28

# LSweek28 - MATH1211/LSweek28/TNK/10-11(2 THE UNIVERSITY OF...

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1 MATH1211/LSweek28/TNK/ /10-11(2) THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211: Multivariable Calculus Lecture Summary – Week 28 March 7, 2011 We first went through briefly the solution to Assignment 3 . We did a quick revision on the procedure of applying the Lagrange Multipliers Method to find extrema of functions subject to constraints. We note that the methods given in Theorems 3.1 and 3.2 (p.261 and p.264) are only for finding critical points, and these critical points are candidates that may or may not give local extrema. To complete the solution, we need to find out whether these critical points actually give local/global minima, or local/global maxima, or not. This last part of the solution may not be easy. We considered Exercise 17 on p.271. Brief solution : We wish to maximize , subject to constraint , where , and . From we get . This, together with the constraint equation give the following system of equations: Solving (1) and (2) we obtain , which is , which implies either or . Since we require , it then follows . Similarly, solving (1) and (3) gives . Substituting into (4) we get . Since (in fact for all ), (6,6,6) is a valid critical point. [Note: the checking of for the points obtained is an important step in using the Lagrange Multipliers method.] (The final checking of (6,6,6) really gives the global maximum of for is left as an exercise.) Alternative solution to Exercise 17 : Since , by the (geometric mean) (arithmetic mean) inequality, we get , with equality holds iff . Now that , it follows that , with equality holds iff . Thus is the unique solution that gives the global maximum of f . However, note that this solution is not using the Lagrange Multipliers method.

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LSweek28 - MATH1211/LSweek28/TNK/10-11(2 THE UNIVERSITY OF...

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