Let $d\geq 4$ and fix $w\in \mathbb{R}.$ Let $$S = \{a = (a_{0},a_{1},\dots ,a_{d})\in \mathbb{R}^{d+1}\mid f_{a}(w) = 0\: \text{and}\: f’_{a}(w) = 0\},$$ where the polynomial function $f_{a}(x)$ is defined as $f_{a}(x):=\displaystyle{}\sum_{i = 0}^{d}a_{i}x^{i}$ and $f’_{a}(w)$ denotes the derivative of $f_{a}(x)$ with respect to $x,$ evaluated at $w.$ 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 $\mathbb{R}^{d+1}$
- None of the other options
