I'm trying to figure out the following problem:
If $\sum_{i=0}^n\frac{a_i}{i+1}=0$, then prove $\sum_{i=0}^n a_i x^i = 0$ has a solution in $(0,1)$.
I'm trying to do the following:
Let $f(x) = \sum_{i=0}^n \frac{a_i x^{i+i}}{i+1} \implies f'(x) = \sum_{i=0}^na_ix^i, f(1) = \sum_{i=0}^n\frac{a_i}{i+1}$. And I want to somehow use Rolle's theorem in reverse. My understanding is that to use Rolle's theorem normally, I would show that $f$ is continuous on $[0,1]$ and differentiable on $(0,1)$ where $f(a) = f(b)$ which then implies there is a solution in $(0,1)$. However, I am starting with assuming there is a solution in $f'$ and trying to show there is a solution in $f$. I feel like I need to somehow use Rolle's in reverse, but I'm not sure how to show this.
You are nearly there: $f(0)=0=f(1)$. So just apply Rolle to $f$.