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Big-O relationship proof

  1. Dec 10, 2012 #1
    1. The problem statement, all variables and given/known data
    To establish a big - O relationship, find witnesses C and k such that |f(x)| <= C|g(x)| whenever x > k.

    Show that x^3 is O(x^4) but that x^4 is not O(x^3).


    2. Relevant equations
    If f(x) is O(g(x)) then there is a C such that f(x) <= Cg(x).


    3. The attempt at a solution

    If f(x) is O(g(x)) then there is a C such that f(x) <= Cg(x).
    Thus, if x^3 is O(x^4), then x^3 <= C(x^4).
    This is true for all C > 0 when C is a real number.

    Similarly, if x^4 is O(x^3) then
    x^4 <= Cx^3.
    x <= C (divide both sides by x^3)
    However, x <= C can not be true for any C. Thus x^4 is not O(x^3).

    I'm not sure about this proof because I didn't provide witnesses C and k as the instructions called for. Big-O problems confuse me and I'm not sure if this proof is correct. Any suggestions are appreciated.
     
  2. jcsd
  3. Dec 11, 2012 #2
    Any suggestions?
     
  4. Dec 11, 2012 #3
    What is true? x3≤Cx4 if and only if x ≥ 1/C. Perhaps this can help you to find some values of k and C.
     
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