Finding the average value of a cubic function 
Let $p(x)=ax^{3}+bx^{2}+cx+d$. There exists real numbers $r$ and $s$(independent of $a,b,c$ and $d$) $0<r<s<1$. For which the average value of $p(x)$ on the interval $[0,1]$ is equal to the average value of $p(r)$ and $p(s)$. Find the product of $rs$ expressed as a fraction. 

What I know is $$f_{\text{avg}}=\frac{1}{b-a}\int\limits_a^bf(x)dx$$
$$f_{\text{avg}}=\int\limits_0^1f(x)dx$$ 
From that I am having trouble finding $r$ or $s$. I would think they would be the same but they clearly can't be. I know how to plug in the right values and take the integral but after that unsure. I'm having trouble understanding the question or what to try. Thanks for any help. 

The question probably misses a condition $a,b,c,d\ne0$.
We have
\begin{align}
p_{\mathrm{avg}} &= \tfrac14 a+\tfrac13 b+\tfrac12 c+d
=p(r)+p(s)
=
\tfrac12 (r^3+s^3)a
+
\tfrac12 (r^2+s^2)b
+
\tfrac12(r+s)c
+d
\end{align}
From this system we have
\begin{align}
\tfrac12 r+\tfrac12s &=\tfrac12
\\
\tfrac12 s^2+\tfrac12 r^2&=\tfrac13
\end{align}
Which gives $rs=\tfrac16$.
Moreover, we can find that 
$r=\tfrac12-\tfrac16\sqrt{3}$,
$s=\tfrac12+\tfrac16\sqrt{3}$.
