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,
- $S$ is finite or infinite depending on the value of $\alpha$
- $S$ is a $2$-dimensional vector subspace of $\mathbb{R}^{d+1}$
- $S$ is a $d$-dimensional vector subspace of $\mathbb{R}^{d+1}$
- $S$ is a $(d-1)$-dimensional vector subspace of $\mathbb{R}^{d+1} $
- 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 $\alpha$
