Answer
Smooth function
Work Step by Step
Step 1: Given \[ \mathbf{r}(t) = t^3\mathbf{i} + (3t^2 - 2t)\mathbf{j} + t^2\mathbf{k} \] Step 2: Since \[ \mathbf{r}'(t) = 3t^2\mathbf{i} + (6t - 2)\mathbf{j} + 2t\mathbf{k} \] The components are continuous functions on \(\mathbb{R}\), and there is no value of \(t\) for which all three of them are zero. It is clear that \(3t^2\), \(6t - 2\), and \(2t\) are continuous functions, and there is no \(t\) that satisfies \(\mathbf{r}'(t) = \mathbf{0}\), hence \(\mathbf{r}'(t) \neq \mathbf{0}\) and \(\mathbf{r}(t)\) is a smooth function.
