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Homework Help: Most likely speed in Maxwell-Boltzmann distribution

  1. Dec 24, 2015 #1
    1. The problem statement, all variables and given/known data
    What is the most likely speed in Maxwell-Boltzamann distribution?

    2. Relevant equations
    [itex] f(v)dv=4\pi(\frac{m}{2 \pi kT})^{3/2}v^2Exp(-\frac{mv^2}{2kT})dv[/itex]

    3. The attempt at a solution
    I know I need maximum of f(v) -> [itex]\frac{df}{dv}=0[/itex]. But it is not trivial to do. I found some solution where they said: [itex]\frac{d}{dv^2}(\ln[v^2Exp(-\frac{mv^2}{2kT})])=0[/itex]. But I don't know how they arrive to it. Could somebody advise?
  2. jcsd
  3. Dec 24, 2015 #2
    Oooou sry, it is easy. I made a mistake I derivate f(v) - function without v2 term. Otherwise, why they can rewrite this problem to logarithm and derivative according to v2.
  4. Dec 24, 2015 #3
    They just felt like its easier to differentiate the natural log of f, rather than f itself. Both f and its natural log have a maximum at the same value of v.
  5. Jan 6, 2016 #4
    Thank you for your response. So can I say generally that if I apply some monotonic function to other, that extremes stay at same point?
  6. Jan 6, 2016 #5
    What are your thoughts on this?

  7. Jan 6, 2016 #6
    What are you asking me now? How or my oppinion if it is true?
  8. Jan 6, 2016 #7
    I'm asking to see if you can reason it out mathematically.
  9. Jan 6, 2016 #8
    No. I just ask :)
  10. Jan 6, 2016 #9
    Suppose g(y) is a monotoically increasing function of y, and y(x) is a function of x with a maximum. Then, by the chain rule,
    dg/dy is always positive, so dg/dx has a zero derivative at the same location where dy/dx has a zero derivative.
  11. Jan 6, 2016 #10
    Very nice ;) It is true what I said.
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