I have this question:
Note: $f$ is assumed to be continuous. screenshot of question
Suppose $\int_{a}^{b} f(x) v(x) d x=0$ for all $v \in C_{0}^{1}(a, b) .$ Show that $f(x)=0$ for all $x \in[a, b]$. What about if $\int_{a}^{b} f(x) v(x) d x=0$ for all $v \in C_{0}^{2}(a, b)$, or for all $v \in C_{0}^{3}(0,1) ?$
I have been trying to figure out an angle of attack but I am completely stumped. First I am a little thrown off by the subscript in $C^2_0$ someone said it means the derivative vanishes at the endpoints but I am not sure.
One thing I wanted to say is, if this applies for all $v$ then it applies for $v$ constant, but all that really tells me is that $F(x) = \int f(x) dx$ is equal at the endpoints.
I don't really want the answer as I want to try to solve this myself, but a hint as to a useful theorem to read, an example or a little trick would be awesome.
