Identify the correct steps involved in the proof of the statement "If (x) and g(x) are functions from the set of real numbers to the set of real numbers, then f(x) is (g(x)) if and only if there are positive constants k, C₁, and C2 such that C₁lg(x) ≤ f(x) s C₂lg(x) whenever x> k." (Check all that apply.) Check All That Apply Suppose 1x) is (g(x)). Then, fx) is both O(g(x)) and (g(x)). There exists positive constants C₁, k₁. C₂. and k2 such that Ix) ≤ C₂lg(x) for all x> k2 and x) Gix) for all x> k₁. Let k=max(k₁, k2). Then, for x>k, G|g(x)| ≤ (x)| ≤ C₂lg(x)|. Let k = min(k₁, k₂). Then, for x>k, Clg(x)| ≤ (x)| ≤ C₂lg(x). Suppose there exists positive constants C₁, C₂, and k such that Gigxsf(x) C₂lg(x) whenever x> k Let k = k₁=k₂. Then, Ixls C₂lg(x) for all x> k₂ and 1x) Gig(x) for all x> k₁. Let k= k₁=k₂. Then, Ixls C₂lg(x) for all x> kand Ix) s Gig(x) for all x>k.
Identify the correct steps involved in the proof of the statement "If (x) and g(x) are functions from the set of real numbers to the set of real numbers, then f(x) is (g(x)) if and only if there are positive constants k, C₁, and C2 such that C₁lg(x) ≤ f(x) s C₂lg(x) whenever x> k." (Check all that apply.) Check All That Apply Suppose 1x) is (g(x)). Then, fx) is both O(g(x)) and (g(x)). There exists positive constants C₁, k₁. C₂. and k2 such that Ix) ≤ C₂lg(x) for all x> k2 and x) Gix) for all x> k₁. Let k=max(k₁, k2). Then, for x>k, G|g(x)| ≤ (x)| ≤ C₂lg(x)|. Let k = min(k₁, k₂). Then, for x>k, Clg(x)| ≤ (x)| ≤ C₂lg(x). Suppose there exists positive constants C₁, C₂, and k such that Gigxsf(x) C₂lg(x) whenever x> k Let k = k₁=k₂. Then, Ixls C₂lg(x) for all x> k₂ and 1x) Gig(x) for all x> k₁. Let k= k₁=k₂. Then, Ixls C₂lg(x) for all x> kand Ix) s Gig(x) for all x>k.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Identify the correct steps involved in the proof of the statement "If f(x) and g(x) are functions from the set of real numbers to the set of real numbers, then f(x) is (g(x)) if and only if there are positive constants k, C₁, and C₂ such that C₁lg(x)| ≤ f(x)| ≤
C₂lg(x) whenever x> k." (Check all that apply.)
Check All That Apply
Suppose f(x) is (g(x)). Then, f(x) is both O(g(x)) and 2(g(x)).
There exists positive constants C₁, k₁, C₂, and k2 such that If(x) ≤ C₂lg(x) for all x> k2 and If(x) ≥ C₁lg(x) for all x> k₁.
Let k = max{k₁, k2}. Then, for x>k, C₁lg(x)| ≤ f(x)| ≤ C₂lg(x).
Let k = min{k₁, k2). Then, for x>k, C₁lg(x)| ≤ f(x)| ≤ C₂lg(x).
Suppose there exists positive constants C₁, C₂, and k such that C₁lg(x)| ≤ f(x)| ≤ C₂lg(x)| whenever x> k.
Let k = k₁=k2. Then, If(x) ≤ C₂lg(x) for all x> k2 and Ix) ≥ C₁lg(x) for all x> k₁.
Let k = k₁=k2. Then, If(x) ≤ C₂lg(x) for all x>k and f(x)| ≤ C₁lg(x)l for all x> k.
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