Given $a_0, a_1, .., a_n$ are the real numbers satisfying $$\dfrac {a_0}{n+1} + \dfrac {a_1}{n} +......+\dfrac {a_{n-1}}{2}+a_n=0$$ then prove that there exists at least one real root of the equation $a_0 x^n + a_1 x^{n-1} +....+a_n=0$ such that $x\in (0,1)$.
My teacher told me to integrate the left hand side of the given equation and call it as $g(x)$, then after verifying the requirements of Rolle's Theorem again differentaite $g(x)$ to get the required equation and this completes the proof. However, I don't understand that first integrating and then differentiating the same thing works here.
Let $$p(x) = a_0x^n + a_1x^{n-1}+\ldots + a_n$$
and let $g$ be an antiderivative of $p$ which satisfies $g(0)=0$, that is
$$g(x) = \frac{a_0x^{n+1}}{n+1}+\frac{a_1x^n}{n}+\ldots +a_nx$$
Check that $g(1)=g(0)=0$, now apply Rolle's theorem to conclude about root of $p$.