# Hint/angle of attack for this calculus of variations problem?

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.

• No it is not necessarily continous Commented Feb 4, 2022 at 6:26
• Sorry I went back and checked, yes f is continuous, my bad, let me edit. Commented Feb 4, 2022 at 6:49
• Hint: suppose $f(x_0) \neq 0$. Continuity allows you to find an interval $I$ around $x_0$ so that the function varies around it by at most $\epsilon$. Consider a cutoff function $v$ supported in that interval...
– user752785
Commented Feb 4, 2022 at 6:52
• Do note that while the result of this question may be important in the calculus of variations, it's not actually a problem about the calculus of variations but rather about a separating set of functionals on $C(a,b).$ Commented Feb 4, 2022 at 8:00
• @TheHype excuse the silly question, what is a cut off function? Commented Feb 4, 2022 at 8:29