Problem: if function $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, and $$\int_a^b f(x)dx=0$$ Prove that there exists two distinct real numbers $\xi_1,\xi_2\in(a,b)$ such that $$f'(\xi_1)(\xi_1-a)+f'(\xi_2)(\xi_2-b)+f(a)+f(b)=0$$
I suspect I should use some sort of mean value theorem to prove this problem, but I tried all forms of the theorem listed on the Wikipedia page without any success. I am beginning to suspect this problem might be wrong and am looking for counter-example. Any help from you is greatly appreciated!
Take $f(x) = x^{10}-1$ on the interval $[0,11^{1/10}]$, it meets all your conditions;
Given that $f(0) +f(11^{1/10}) = 9$ and $f'(x) = 10x^9$ we are trying to find $\alpha,\beta \in (0,11^{1/10})$ such that
$$10\alpha^{10}+10\beta^9(\beta-11^{1/10}) +9 = 0 $$
but notice that $10\beta^9(\beta-11^{1/10}) +9$ is always positive (and also $10\alpha^{10}$ obviously) so the theorem is not true!
The way I found it :
I started to reason about $\cos x$ and $\arcsin x$ but then I realized that this "type" of functions attained the same derivative two times and they were also too symmetrical. Therefore I tried $x^2-1$ to avoid the problem but it wasn't strong enough (one of the two parts in which I also decomposed the expression resulting from $x^{10}-1$ was negative) then I tried $\alpha x^k + \beta$ and there were many counter-examples :-)