mathCalculus4-07-draft - ARE211, Fall 2007 LECTURE #19:...

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Preliminary draft only: please check for Fnal version ARE211, Fall 2007 LECTURE #19: THU, NOV 1, 2007 PRINT DATE: AUGUST 21, 2007 (CALCULUS4) Contents 4. Univariate and Multivariate Diferentiation (cont) 1 4.6. Taylor’s Theorem (cont) 1 4.7. Application oF Taylor’s theorem: second order conditions For an unconstrained maximum. 2 4.8. Another application oF Taylor 4 4.9. Terminology Review 5 4. Univariate and Multivariate Differentiation (cont) 4.6. Taylor’s Theorem (cont) Taylor’s Theorem (continued): Why is the theorem so tremendously important? Because iF you are only interested in the sign oF ( f x + dx ) - f x )) and you have an n ’th order Taylor expansion, then you know that For some neighborhood about ¯ x , the sign oF your expansion will be the same as the sign oF the true diference. 1
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2 LECTURE #19: THU, NOV 1, 2007 PRINT DATE: AUGUST 21, 2007 (CALCULUS4) 4.7. Application of Taylor’s theorem: second order conditions for an unconstrained maximum. Going to be talking about necessary and sufcient conditions For an optimum oF a di±erentiable Function. Terminology is that ²rst order conditions are necessary while second order conditions are sufcient . The terms necessary and sufcient conditions have a Formal meaning: IF an event A cannot happen unless an event B happens, then B is said to be a necessary condition For A . IF an event B implies that an event A will happen, then B is said to be a sufcient condition For A . ³or example, consider a di±erentiable Function From R 1 to R 1 . f cannot attain an interior maximum at ¯ x unless f p x ) = 0. i.e., the maxmimum is A ; the derivative condition is B . Thus, the condition that the ²rst derivative is zero is necessary For an interior maxi- mum; called the ²rst order conditions. Emphasize strongly that this necessity business is delicate: derivative condition is only necessary provided that f is di±erentiable and we’re talking interior maximum Also, only talking LOCAL maximum. f p x ) = 0 certainly doesn’t IMPLY that f attains an interior maximum at ¯ x IF f pp x ) < 0, then the condition f p x ) = 0 is both necessary and sufcient For an interior local maximum; Alternatively, iF you know in advance that f is strictly concave , then the condition that f p x ) is zero is necessary and sufcient For a strict global maximum.
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ARE211, Fall 2007 3 Generalizing to functions deFned on R n , a simple application of Taylor’s theorem proves that if
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.

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mathCalculus4-07-draft - ARE211, Fall 2007 LECTURE #19:...

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