Find the smallest constant $C$ such that for every polynomial $P(x)$ of degree $3$ with a root in $[0,1]$,
$$\int_0^1 |P(x)|dx\leq C\max_{x\in[0,1]}|P(x)|.$$
Here's my rough work. Write $P(x)=a_3x^3+a_2x^2+a_1x+a_0$. Then $$\int_0^1 P(x)dx=\dfrac{a_3x^4}{4}+\dfrac{a_2x^3}{3}+\dfrac{a_1x^2}{2}+a_0x.$$ I think I'm supposed to assume something about $P(x)$ to make the problem easier. That might be that $\max_{x\in[0,1]} P(x)=1$. I'm not sure if I can assume that $P(x)$ is always positive over the interval. Also, for the case that $P(x)$ has $3$ roots in the interval $[0,1]$, I can split the interval into the intervals $[0,I_1], [I_1,I_2],[I_2,I_3]$, where $I_1,I_2,I_3$ are the roots of $P(x)$. I suppose I can also assume $P(x)\geq 0$ on $[0,1]$?