tmua.me Tau Paper 2 · Question 18

Let $\mathcal P$ be the set of real polynomials $p$ of degree at most $6$ such that

p(x)\ge0\quad(-1\le x\le1),\qquad \int_{-1}^1p(x)\,\mathrm dx=1.

Write $a_6(p)$ for the coefficient of $x^6$ , taking $a_6(p)=0$ when the degree is less than $6$ . Define

A=\max_{p\in\mathcal P}p(0),\qquad B=\max_{\substack{p\in\mathcal P\\a_6(p)\ge0}}p(0).

Let $N$ be the number of polynomials attaining the second maximum which are non-negative for every real $x$ . Which ordered triple is $(A,B,N)$ ?

\displaystyle \left(\frac{45}{32},\frac98,1\right)

\displaystyle \left(\frac{45}{32},\frac98,\infty\right)

\displaystyle \left(\frac{45}{32},\frac54,0\right)

\displaystyle \left(\frac{45}{32},\frac{45}{32},1\right)

\displaystyle \left(\frac98,\frac98,1\right)

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