Application Of Derivatives

Consider the following statements

Statement I If $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\ldots .+\frac{a_n}{n+1}=0$, where $a_0, a_1, \ldots, a_n$ are real numbers, then the polynomial $a_0+a_1 x+a_2 x^2+\ldots .+a_n x^n$ has a zero in the interval $(0,1)$.

Statement II If $f:[a, b] \rightarrow \mathbf{R}$ is continuous on $[a, b]$ and $f$ is differentiable in $(a, b)$, where $a>0$ and if $\frac{f(a)}{a}=\frac{f(b)}{b}$, then there exists $c \in(a, b)$, such that $c f^{\prime}(c)=f(c)$.

Which one of the following options is true?