Given $a_0, a_1,…,a_n$ are the real numbers satisfying

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.

• Rolle's Theorem allows you to conclude something about a value of $f'(x)$ given some information about some values of $f(x)$. You want to conclude something about $a_0x^n+a_1x^{n-1}+\cdots +a_n$, so you need to view this as $f'(x)$ for some $f(x)$. Hence you need to integrate to figure out what $f(x)$ is. – kccu Mar 31 at 17:39

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 $$p$$.