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mathCalculus4-07-draft

# mathCalculus4-07-draft - ARE211 Fall 2007 LECTURE#19 THU...

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Preliminary draft only: please check for final version ARE211, Fall 2007 LECTURE #19: THU, NOV 1, 2007 PRINT DATE: AUGUST 21, 2007 (CALCULUS4) 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 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 difference. 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 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 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.
ARE211, Fall 2007 3 Generalizing to functions defined on R n , a simple application of Taylor’s theorem proves that if

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mathCalculus4-07-draft - ARE211 Fall 2007 LECTURE#19 THU...

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