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Unformatted text preview: Lecture 9 Recall: L = [ a,b ] ⊂ U F → Y F differentiable at each point of L Ψ ∈ Y * ⇒ ∃ p ∈ L such that Ψ( F ( b ) F ( a )) = Ψ( F p ( b a )) Theorem 1 (MVT) . If F : U → Y is differentiable at each point of [ a,b ] ⊂ U then there exists p ∈ ( a,b ) such that k F ( b ) F ( a ) k ≤ k F p ( b a ) k ( ≤ k F p kk b a k ) Proof. Let Ψ ∈ Y * be such that k Ψ k = 1 and Ψ( F ( b ) F ( a )) = k F ( b ) F ( a ) k . Apply previous MVT: there exists p ∈ ( a,b ) such that k F ( b ) F ( a ) k = Ψ( F ( b ) F ( a )) = Ψ( F p ( b a )) ≤ k Ψ kk F p ( b a ) k = k F p ( b a ) k ( k Ψ k = 1) . Theorem 2 (MVT) . Let F : U → Y be differentiable and [ a,b ] ⊂ U . If p ∈ U then k F ( b ) F ( a ) F p ( b a ) k ≤ k b a k sup q ∈ [ a,b ] k F q F p k . 1 Proof. Apply the previous MVT to G : U → Y : x 7→ F ( x ) F p ( x ) . G is differentiable at each q ∈ U and G q ( h ) = F q ( h ) F p ( h ) ....
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This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.
 Summer '10
 Staff
 Algebra

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