Solutions

Let

$$f(x)=a_0x+\frac{a_1x^2}{2}+\cdots+\frac{a_nx^{n+1}}{n+1}.$$

Then

$$f'(x)=a_0+a_1x+\cdots+a_nx^n$$

and by the mean value theorem,

$$\frac{f(1)-f(0)}{1-0}=a_0+\frac{a_1}{2}+\cdots+\frac{a_n}{n+1}=f'(c)$$

for some $c\in(0,1)$. But since

$$\sum_{i=0}^n \frac{a_i}{i+1}=0$$

then

$$f'(c)=a_0+a_1c+\cdots+a_nc^n=0.$$