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There is a problem which I have been thinking about but can't come to a conclusion. And Final exam is tomorrow: enter image description here

My attempt: To find solutions to both maximization and minimization problems before even solving it requires the set to be compact and continuous by the weierstrass theorem (aka extreme value theorem). Well, then we are supposed to show the constraint sets are closed and bounded, first, to determine whether the sets are compact. S={(x,y,z)∈R^3|x^2+y^2=4} is a bounded set, because it represents a circle of radius r=2 centered at the origin. What about x-z=1? it seems like it's an infinite line and thus no conceivable finite ball could entirely contain it. So not bounded? How do you determine boundedness of the constraint sets? Or whether they are closed? and continuous? Ways to tell them generally? Thanks a lot in advance.

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    $\begingroup$ what you have is the intersection of a plane and a cylinder which is an ellipse $\endgroup$
    – janmarqz
    Jan 7 at 1:54

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The set $S$ is not bounded, because its definition does not depend on the value of $z$. Also, $x-z=1$ is a plane, not a line.

Maybe the problem is telling you that both constraints must be satisfied simultaneously, so you are constrained to the intersection of $S$ and the plane $x-z=1$, which is a compact set.

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