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## Homework Statement

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

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

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HallsofIvy

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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.

- #4

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As you said,

[tex]f=e^{-ax}[/tex]

[tex]f'=-ae^{-ax} [/tex] This does not change signs(a is const),its monotonous.

[tex]f=e^{-ax}[/tex]

[tex]f'=-ae^{-ax} [/tex] This does not change signs(a is const),its monotonous.

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- #6

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Read again, I said [itex] \ f' [/itex] is not changing signs.you pointed out that this does not change sign, how?

[itex] f'' \mbox{and} f' \mbox{are both having same sign for}\ a<0[/itex]the function is positive and its derivative is negative and the second derivative will agian be positive.

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