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Which is larger?

  1. Nov 23, 2009 #1
    1. The problem statement, all variables and given/known data
    which value is larger?

    1001^(1/1001) or 1002^(1/1002)


    2. Relevant equations
    ???


    3. The attempt at a solution

    Honestly, I am pretty stuck on what to do here. We can use calculus to prove this (might be needed?) I cannot use a calculator, have to make it obvious that one is larger than the other. Where would i start with this?
     
  2. jcsd
  3. Nov 23, 2009 #2

    Mark44

    Staff: Mentor

    Look at the function x1/x, and in particular, its derivative. If the derivative is negative, the function is increasing. If positive, the function is increasing. One of these should tell you something about the inequality you are investigating.

    Be careful! The derivative of x1/x is NOT (1/x)x1/x - 1.
     
  4. Nov 23, 2009 #3
    So derivative of x^1/x is x^(1/x)*ln(x). I am doing it for all x >1, so that would mean that this function is increasing.

    I can also do x+1^(1/(x+1)) and the derivative of that, but where can I go from there? Now I just have two derivatives that are no obvious than the original question...
     
  5. Nov 24, 2009 #4

    Mark44

    Staff: Mentor

    That's not the derivative.

    If x > 0, x = eln x, right? So x1/x = (eln x)1/x = e(1/x)*ln x. Now differentiate.
     
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