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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. Taylors Theorem (cont) 1 4.7. Application of Taylors 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. Taylors Theorem (cont) Taylors 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 Taylors 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 were talking interior maximum Also, only talking LOCAL maximum. f prime ( x ) = 0 certainly doesnt 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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 Fall '07
 Simon

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