I'm given this 2-D curve: $\gamma(t)=(2cos(t),3sin(t))$ where $t \in [0,2\pi]$. I want to find the points of max and min curvature using calculus, but I keep second guessing my approach... maybe I calculated something wrong and I'm just trying to work with a curvature values that is completely from outer space.
Here is what I've done so far:
I plotted the curve on GeoGebra, and found that it is an ellipse.
I calculated the curvature using $\kappa=\frac{|\dot{\gamma}\times\ddot{\gamma}|}{|\dot{\gamma}|^3}$, where the dot notation is just the derivatives of the curve (i.e., $\dot{\gamma}= \frac{d}{dt}\gamma(t)=T(t)v(t)$, $T$ being the tangent vector, $v$ being velocity).
After some lengthy steps, I got that $\kappa=\frac{6}{8\sin^2(t)\sqrt{\sin^2(t)}+27\cos^2(t)\sqrt{cos^2(t)}}$.
Now I know that the min occurs when denominator is large and that the max occurs when the denominator is small. But, I am not sure if I should derive $\kappa$ and solve $\kappa^{\prime}(t)=0$, and then find $\kappa^{\prime\prime}$ and plug in my value for $t$ into $\kappa^{\prime\prime}$, or if I should just plug in values of $t$ straight into $\kappa$ and see what gives me the smallest and largest $\kappa$ values.
The second approach seems intuitively correct, but the first approach seems more in tune with what I remember from calculus.... Or perhaps I calculated something wrong because I really want to simply the denominator but it doesn't seem like it can be simplified. Any thoughts on where I'm screwing up?