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Lecture09 - Lecture 9 Recall F L =[a b U Y F dierentiable...

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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 0 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 0 p ( b - a ) k ( ≤ k F 0 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 0 p ( b - a )) k Ψ kk F 0 p ( b - a ) k = k F 0 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 0 p ( b - a ) k ≤ k b - a k sup q [ a,b ] k F 0 q - F 0 p k . 1
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Proof. Apply the previous MVT to G : U Y : x 7→ F ( x ) - F 0 p ( x ) . G is differentiable at each q U and G 0 q ( h ) = F 0 q ( h ) - F 0 p ( h ) . There exists q [ a, b ] such that k G ( b ) - G ( a ) k ≤ k G 0 q ( b - a ) k . Hence k ( F ( b ) - F 0 p ( b )) - ( F ( a ) - F 0 p ( a )) k ≤ k ( F 0 q - F 0 p )( b - a ) k therefore k F ( b ) - F ( a ) - F 0 p ( b - a ) k ≤ k b - a kk F 0 q - F 0 p k .
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