Question

Let f be a continuous and bounded function on [a,b] such that if

Answer 1

any idea?

Answer 2

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

Answer 3

I guess so..

Answer 4

F'(x)=f(x)

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

Answer 5

that is what i was thinking, yes

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

well then i guess this is the right approach!