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Homework Help: Changing sign in derivatives

  1. Mar 23, 2007 #1
    1. The problem statement, all variables and given/known data

    This is ralated to completely monotone functions. if a function and its successive derivative changes sign then what does this mean?

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 23, 2007 #2

    HallsofIvy

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    Changes sign where? If a function, f, is monotone increasing then its derivative is always positive, but f may be positive for some values of x and negative for others. the second derivative then may be postive for some values of x and negative for others.
     
  4. Mar 24, 2007 #3
    changes sign at every point on the domain.
    if we have a fucntion f(x)=exp(-a*x) then f'(x)=-a*exp(-a*x),
    f''(x)=a^2*exp(-a*x) and so on.....
    similarly if we have f(x)=(a^2+x)^(-1/2) then f'(x)=(-1/2)*(a^2+x)^(-3/2)
    and f''(x)(-1/2)*(-3/2)*(a^2+x)^(-5/2) and so on....
    i think that if we have such kind of functions then there will be successively concave up( the slope will become steeper) and concave down( the slope will be come shallower). i want such kind of explaination.
     
  5. Mar 24, 2007 #4
    As you said,
    [tex]f=e^{-ax}[/tex]
    [tex]f'=-ae^{-ax} [/tex] This does not change signs(a is const),its monotonous.
     
    Last edited: Mar 24, 2007
  6. Mar 24, 2007 #5
    I know that such kind of functions are completely monotone functions. but its successive derivative changes sign. and you pointed out that this does not change sign, how? the function is positive and its derivative is negative and the second derivative will agian be positive.
     
  7. Mar 25, 2007 #6
    Read again, I said [itex] \ f' [/itex] is not changing signs.

    [itex] f'' \mbox{and} f' \mbox{are both having same sign for}\ a<0[/itex]
     
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