QuestionExercises - at which point on $y=a x^{2}+b x+c$ is the curvature maximized? - at which points on the ellipse $\left\{\begin{array}{l}x=a \cos t \\ y=b \sin t\end{array}\right.$ are curvatures maximized/minimized?

Studdy Solution
The expression $a^2 \sin^2 t + b^2 \cos^2 t$ is minimized when $t = 0$ or $t = \pi$, and maximized when $t = \frac{\pi}{2}$ or $t = \frac{3\pi}{2}$.
Thus, the curvature is maximized at $t = \frac{\pi}{2}$ and $t = \frac{3\pi}{2}$, and minimized at $t = 0$ and $t = \pi$.
The point on the parabola where curvature is maximized is $x = -\frac{b}{2a}$. For the ellipse, curvature is maximized at $t = \frac{\pi}{2}, \frac{3\pi}{2}$ and minimized at $t = 0, \pi$.