Find the smallest constant $C$

by ford jones   Last Updated October 20, 2019 05:20 AM - source

Find the smallest constant $C$ such that for every polynomial $P(x)$ of degree $3$ that has a root in $[0,1]$,

$\int_0^1 \vert P(x)\vert dx\leq C\max_{x\in[0,1]}\vert P(x)\vert$.

Here's my rough work. Obviously if $P(x)$ is of degree $3$ then it can be written in the form $ax^3+bx^2+cx+d$. Then $\int_0^1 P(x)dx=\dfrac{ax^4}{4}+\dfrac{bx^3}{3}+\dfrac{cx^2}{2}+dx$. 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 $[0,1]$?

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