Find the curvature of the curve $\overrightarrow r(t)=2t \overrightarrow i +t^2\overrightarrow j + \frac{t^3}{3}\overrightarrow k$. At what point on that curve is curvature maximum?
I found the curvature to be:
$\frac{6}{(t^2+2)^3}$
From here, how do I find the curvature maximum? I assumed I would set the curvature equal to $0$ but that doesn't do anything.
If $\kappa(t) = \frac{6}{(t^2+2)^3}$, we want to find $t$ where $\kappa(t)$ takes a maximum value. We want to set the $derivative$ equal to zero. We get:
$$ \kappa '(t) = \frac{-36t}{(t^2+2)^4} = 0$$
This is only true when $t=0$.