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Let $\mathbb{R}$ denote the set of real numbers. Let $d \geq 4$ and $\alpha \in \mathbb{R}$. Let

$$S=\left\{\left(a_0, a_1, \ldots, a_d\right) \in \mathbb{R}^{d+1}: \sum_{i=0}^d a_i \alpha^i=0 \text { and } \sum_{i=0}^d i a_i \alpha^{i-1}=0\right\} \text {. }$$

Then,

1. $S$ is finite or infinite depending on the value of $\alpha$
2. $S$ is a $2$-dimensional vector subspace of $\mathbb{R}^{d+1}$
3. $S$ is a $d$-dimensional vector subspace of $\mathbb{R}^{d+1}$
4. $S$ is a $(d-1)$-dimensional vector subspace of $\mathbb{R}^{d+1}$
5. For each $\left(a_0, a_1, \ldots, a_d\right) \in S$, the function

$$x \mapsto \sum_{i=0}^d a_i x^i$$

has a local optimum at $\alpha$

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