Exercise of analysis I have to prove that if $F\in AC([a,b])$ then:
$$
\int_a^bF'\varphi dt=-\int_a^bF\varphi' dt  
$$
$\forall\varphi\in C_0^\infty((a,b))$. And show that if $f\in BV([a,b])$ then the equation is not true, in general.
This is an exercise of my analysis course but I don't know how to start. 
Edit: probably a stupid question: but is it obvious that $F\varphi\in AC$?
 A: Since $F$ is absolutely continuous, it is differentiable a.e on $(a,b)$. So, a.e. $(F\varphi)'=F'\varphi+F\varphi'$. This equality will survive integration over $(a,b)$. As $\varphi(b)=\varphi(a)=0$, the left-hand side integrates to zero, and then you get the equality you are looking for (what we are doing is basically verifying that we can integrate by parts). 
As for functions of bounded variation, you can try taking $f$ to be the Cantor function, and $\varphi$ a function such that $f\varphi'$ has nonzero integral. 
A: If $f\in AC$ then $\varphi f\in AC$, and 
$$
0=(\varphi F)(b)-(\varphi F)(a)=0-0=\int_a^b (\varphi F)'=\int_a^b \varphi' F\,dx+\int_a^b \varphi\, F'\,dx,
$$
as for every $g\in AC$, $g$ is a.e. differentiable and $\int_c^d g\,dx=g(d)-g(c)$, for all $c,d$.
A: Hints (for first part):
(1): For absolutely continuous functions $G$, it is true that
$$
G(b) - G(a) = \int_a^b G'(t) \,dt.
$$
(2): $F\varphi$ is absolutely continuous.
(3): absolutely continuous functions are differentiable a.e.
