Changing sign in derivatives

  • Thread starter aquarian11
  • Start date
  • #1

Homework Statement



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

Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,833
962
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.
 
  • #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.
 
  • #4
182
0
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:
  • #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.
 
  • #6
182
0
you pointed out that this does not change sign, how?
Read again, I said [itex] \ f' [/itex] is not changing signs.

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

Related Threads on Changing sign in derivatives

Replies
12
Views
2K
Replies
10
Views
1K
  • Last Post
Replies
8
Views
884
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
7
Views
4K
  • Last Post
Replies
3
Views
2K
Replies
1
Views
808
  • Last Post
Replies
1
Views
297
Top