Changing sign in derivatives

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

 

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.

 

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.