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Let $S$ be the surface in $\mathbb R^3$ given by the equation $z=\frac12 x^2+\frac12 y^2$ and let $R\subset R^3$ be the surface given by $y^2+z^2=3$ ($R$ is the boundary of the infinite cylinder around the $x$-axis). Let $C$ be the closed curve defined as the intersection between $S$ and $R$. I want to find the maximum and minimum of $f(x,y,z)=x^2+y^2+(z-1)^2$ on $C$.

How can I do this? I've tried using the Lagrange multipliers, but it is quite a lot of work.

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    Can you use the fact that, on the surface $S$, $2z = x^2+y^2$? Then, on the surface, $f(x,y,z) = z+(z-1)^2$, which you can easily maximize wrt $z$. You still need a way to use the second constraint, but I'm not quite sure what the second constraint is. What do you mean by "the surface given by $y^2+z^2$"?2012-07-28
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    I made a typo, thank you for pointing it out2012-07-28
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    If you succeed in finding a parametrization $c(t)$ of the intersection curve, then $f\circ c$ is a function $\mathbb{R}\rightarrow \mathbb{R}$.2012-07-28

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