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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