## 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. satellite73

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. satellite73

that is what i was thinking, yes

7. satellite73

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. satellite73

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

11. satellite73

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. satellite73

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

13. satellite73

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. satellite73

well then i guess this is the right approach!

16. cinar