## cinar 3 years ago Let f be a continuous and bounded function on [a,b] such that if

1. cinar

2. cinar

any idea?

3. anonymous

not really,but it kind of looks like a set up for integrating by parts

4. cinar

I guess so..

5. cinar

F'(x)=f(x) so we can say F'(t)=f(t) right..

6. anonymous

that is what i was thinking, yes

7. anonymous

maybe not exactly what you want, but you would get something like $$gF-\int f g'$$

8. cinar

$\lim_{b \rightarrow} \int\limits_{0}^{b }G(t)F'(t)dt=$

9. cinar

I guess enough to show that $\int\limits f g'$ is convergent

10. anonymous

first part is no problem, since $$\lim_{t\to \infty}g(t)=0$$ and $$\int_a^\infty f(t)$$ is bounded

11. anonymous

yeah you hit the nail on the head, and frankly i don't see why that is true, so this might be the wrong approach

12. anonymous

because all you know about $$g'$$ is that $$g'\leq 0$$

13. anonymous

does this come in a section after some theorem or lemma that maybe you are supposed to use?

14. cinar

no, this is homework, but prof gave the hint and said that you should use integration by part..

15. anonymous

well then i guess this is the right approach!

16. cinar