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Unformatted text preview: ARE211, Fall 2007 CALCULUS4: THU, NOV 1, 2007 PRINTED: NOVEMBER 8, 2007 (LEC# 19) Contents 4. Univariate and Multivariate Differentiation (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 6 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 difference. 1 2 CALCULUS4: THU, NOV 1, 2007 PRINTED: NOVEMBER 8, 2007 (LEC# 19) 4.7. Application of Taylor’s theorem: second order conditions for an unconstrained maximum. Going to be talking about necessary and sufficient conditions for an optimum of a differentiable function. Terminology is that first order conditions are necessary while second order conditions are sufficient . The terms necessary and sufficient 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 sufficient condition for A . For example, consider a differentiable function from R 1 to R 1 . • f cannot attain an interior maximum at ¯ x unless f prime (¯ x ) = 0. – i.e., the maxmimum is A ; the derivative condition is B . – Thus, the condition that the first derivative is zero is necessary for an interior maxi mum; called the first order conditions. – Emphasize strongly that this necessity business is delicate: derivative condition is only necessary provided that f is differentiable and we’re talking interior maximum Also, only talking LOCAL maximum. • f prime (¯ x ) = 0 certainly doesn’t IMPLY that f attains an interior maximum at ¯ x ARE211, Fall 2007 3 • If f primeprime (¯ x ) < 0, then the condition f prime (¯ x ) = 0 is both necessary and sufficient for an interior local maximum; • Alternatively, if you know in advance that f is strictly concave , then the condition that f prime (¯ x ) is zero is necessary and sufficient for a strict global maximum....
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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 Berkeley.
 Fall '07
 Simon

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