tmua.me Tau Paper 2 · Question 9

Let $P(x)=x^3-3x$ . For $t\in\{1,2,3\}$ and a non-negative integer $d$ , consider the following claim about every real polynomial $G$ of degree at most $d$ , including the zero polynomial:

If $G'(1)=0$ and $G$ takes the same value at all real roots of $P(x)=u$ , separately for each $u\in\{-2,0,t\}$ , then

P(x)=P(y)\ \Longrightarrow\ G(x)=G(y) \qquad\text{for every real }x,y.

The common values for different $u$ are allowed to differ. A level with just one real root imposes no restriction on $G$ .

For each $t\in\{1,2,3\}$ , let $d_t$ be the greatest $d$ for which this claim is true. What is $(d_1,d_2,d_3)$ ?

(10,8,7)

(8,6,5)

(9,7,6)

(9,8,6)

(9,7,8)

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tmua.me Tau Paper 2 · Q9