Let $S$ be the graph of the function $\,f(x,y) = 2x^2+ y^2$ in $\,\mathbb{R}^3$ (a paraboloid with vertex at the origin). It is clear that $S$ is a regular surface which can be parametrized by the function $\,\boldsymbol{x} : \mathbb{R}^2 \to S\,$ given by $\,\boldsymbol{x}\left(u,v\right) = \left(u, v, 2u^2 + v^2\right)$. Out of all the differentiable parametrized curves $\,\alpha : \left(- \varepsilon, \varepsilon\right) \to S\,$ such that $\,\alpha \left(0\right) = \left(0,0,0\right),\,$ find one with minimum curvature. Does there exist such a curve with maximum curvature?

This is a homework problem that talks about maximizing or minimizing all the possible values of the curvature of a smooth curve in the paraboloid that goes through the origin and not about the minimum or maximimum normal curvatures which is what confuses me. I guess that the question should still somehow be related to the normal curvature of a curve, but I'm kinda lost on how to attack the problem to be honest, including the question about the existence of a curve with maximum curvature. Any hints or ideas would be greatly appreciated. Thanks for all your answers and comments in advance.


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