Can we use the Second Mean Value Theorem over infinite intervals? Let $[a,b]$ be any closed interval and let $f,g$ be continuous on $[a,b]$ with $g(x)\geq 0$ for all $x\in[a,b]$. Then the Second Mean Value Theorem says that
$$\int_a^bf(t)g(t)\text{d}t = f(c)\int_a^b g(t)\text{d}t,$$
for some $c\in(a,b)$.
Does this theorem work on the interval $[0,\infty]$ ?
EDIT: Assuming the integrals involved converge.

I believe you mistated the second mean value theorem, what you stated was the genearlized form of the first mean value theorem.
The second mean value theorem states:
Let g(x) be a monotonic function over (a,b), and let f(x) not change signs more than a finite number of times over (a,b). Then
∫ b a f(t)g(t)dt=f(a)∫ e a g(t)dt + f(b)∫ b e g(t)dt 
for some e in (a,b)
And under certain conditions this would hold in an infinite interval as well.
