Recall that Taylor's Remainder Theorem states: For any a,b∈R, exists z∈[a,b], such that f(b)=f(a)+f′(a)(b−a)+21​f′′(z)(b

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Recall That Taylor S Remainder Theorem States For Any A B R Exists Z A B Such That F B F A F A B A 21 F Z B 1 (52.22 KiB) Viewed 47 times

Recall that Taylor's Remainder Theorem states: For any a,b∈R, exists z∈[a,b], such that f(b)=f(a)+f′(a)(b−a)+21​f′′(z)(b−a)2. Assuming that there exists L≥0 such that for all a,b∈R,∣f′(a)−f′(b)∣≤L∣a−b∣, prove the following statement: For any x∈R, there exists some η>0, such that if xˉ:=x−ηf′(x), then f(xˉ)≤f(x), with equality if and only if f′(x)=0. (Hint: first show that the assumption implies that f has bounded second derivative, i.e., f′′(z)≤L (for all z ); then apply the remainder theorem and analyze the difference f(x)− f(xˉ)).