Question #41139

Let $L_1$ and $L_2$ be two lines in $\mathbb{R}^3$ given by: $L_1: \vec{r} = \lambda(\hat{i} + 2\hat{j})$ $L_2: \vec{r} = (\hat{i} - 2\hat{j} + \hat{k}) + \mu(2\hat{j} + \hat{k})$ Let $d$ be the shortest distance between $L_1$ and $L_2$. Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = x^3 - 3dx^2 + 3(3d-2)x$. Consider the vectors $\vec{u} = x\hat{i} + \hat{j} + \hat{k}$, $\vec{v} = \hat{i} + x\hat{j} + \hat{k}$, and $\vec{w} = \hat{i} + \hat{j} + x\hat{k}$. A function $g: \mathbb{R} \to \mathbb{R}$ is defined as: $g(x) = \begin{cases} \frac{[\vec{u} \ \vec{v} \ \vec{w}]}{|\vec{u} \times \vec{v}|^2} & \text{if } \vec{u} \text{ and } \vec{v} \text{ are non-collinear} \\ k & \text{if } \vec{u} \text{ and } \vec{v} \text{ are collinear} \end{cases}$ If $g(x)$ is continuous for all $x \in \mathbb{R}$, which of the following is true?