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Question Number 166805 by mathocean1 last updated on 28/Feb/22
f(x)=(x/(1+∣x∣)). show that ∃ K ∈ R_+ such that ∀ x, y ∈ R, ∣f(x)−f(y)∣≤K∣x−y∣
f(x)=x1+∣x∣.showthat∃K∈R+suchthat∀x,y∈R,∣f(x)−f(y)∣⩽K∣x−y∣
Answered by TheSupreme last updated on 28/Feb/22
f′(x)=((1+∣x∣−∣x∣)/((1+∣x∣)^2 ))=(1/((1+∣x∣)^2 ))≤1 f(x)−f(x+h)≤f(x)−(f(x+h)+h)≤−h f(x)−f(y)≤x−y ((f(x)−f(y))/(x−y))≤1
f′(x)=1+∣x∣−∣x∣(1+∣x∣)2=1(1+∣x∣)2⩽1f(x)−f(x+h)⩽f(x)−(f(x+h)+h)⩽−hf(x)−f(y)⩽x−yf(x)−f(y)x−y⩽1

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