Question #41370

Let $\vec{a} = t \hat{i} + t^2 \hat{j} + \hat{k}$, $\vec{b} = 2 \hat{i} + 4 \hat{j} + \hat{k}$, and $\vec{c} = 4 \hat{i} + 16 \hat{j} + \hat{k}$ be three vectors parameterized by $t \in \mathbb{R}$. Let $\vec{p} = \vec{a} \times \vec{b}$, $\vec{q} = \vec{b} \times \vec{c}$, and $\vec{r} = \vec{c} \times \vec{a}$. For $t \in (0, 6)$ such that $[\vec{p} \, \vec{q} \, \vec{r}] \neq 0$, let $\vec{p}', \vec{q}', \vec{r}'$ be the reciprocal system of vectors for $\vec{p}, \vec{q}, \vec{r}$. If $f(t)$ is the continuous extension of the function $g(t) = \frac{1}{2 \sqrt{[\vec{p}' \, \vec{q}' \, \vec{r}']}}$ over the interval $(0, 6)$, find the sum of all values of $t$ at which $f(t)$ attains a local extremum.