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 $P(x) = a_0+a_1 x+a_2 x^2+\ldots+a_n x^n$ has a zero in the interval $(0,1)$.
Statement $II$: If $f:[a, b] \rightarrow R$ 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?

• A
  Only $I$ is true
• B
  Only $II$ is true
• C
  Neither $(I)$ nor $(II)$ is true
• D
  Both $(I)$ and $(II)$ are true