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Unformatted text preview: Supplemental Lecture 2 Optimization in One Dimension II  2 Differentiability in One Dimension • f is differentiable at x ∈ D iff exists. • If f is differentiable at x , L is the derivative of f at x , denoted f ′ ( x ) or df ( x )/ dx . • Differentiability implies continuity but not vice versa. x x f x x f L x ∆ ∆ + = → ∆ ) ( ) ( lim II  3 MeanValue Theorem • Let f : [ a , b ] → R be a differentiable function. • Then there must exist c ∈ ( a , b ) such that • f ′ ( x ) 0 for all x iff f is weakly • If f ′ ( x ) 0 for all x , f must be strictly . ) ( ) ( ) ( a b a f b f dx c df = ≥ ≤ increasing. decreasing. > < increasing. decreasing. a b c II  4 Global Minima and Maxima • Let x * ∈ D . • x * is a global maximum of f iff, for all x ∈ D , f ( x ) ≤ f ( x *). • Likewise, x * is a global minimum of f iff, for all x ∈ D , f ( x ) ≥ f ( x *). • When optimizing the function f on D , our aim is to find a global maximum (or minimum) of f . II  5 Local Minima and Maxima • Let x * ∈ D . • x * is a local maximum of f iff there exists a δ > 0 such that for all x ∈ D ∩ ( x *  δ , x * + δ ), f ( x ) ≤ f ( x *). • Likewise, x * is a local minimum of f iff there exists a δ > 0 such that for all x ∈ D ∩ ( x *  δ , x * + δ ), f ( x ) ≥ f ( x *). II  6 Open Sets • A set S ⊆ R is open if any point in S is a member of an open interval that is also contained in S . • Formally, for any x ∈ S , there exists ε > 0 such that ( x ε , x + ε ) ⊆ S. II  7 FirstDerivative Test • If f : D → R is differentiable, where D is open, and x * ∈ D is a local maximum of f then f ′ ( x *) = 0. • The first step to find a maximum of a differentiable function f is solve for roots of f ′ ....
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This note was uploaded on 01/09/2011 for the course ECON 7230 taught by Professor Feigenbaum during the Spring '10 term at Utah Valley University.
 Spring '10
 Feigenbaum
 Macroeconomics

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