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I need to find the maximum of the function $\ f(x,y,z) = y $ on the following closed set :

$\ y^2+x^2 + z^2 = 3 $

$\ y+ x + z=1$

But I don't have a clue on how to do it ...

Trivially i can get that

$\ 0 < \max(f) < \sqrt{3} $.

Can you give me some pieces of advice on how to solve this kind of problem .

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Are Lagrange Multipliers a tool you can use? More simply one can use the geometry. –  André Nicolas Dec 18 '11 at 18:29
    
no i cant use lagrange multipliers... –  clemclem Dec 18 '11 at 18:31
    
your trivial is not correct because $3^{1/2}$ does not satifies in $y+x+z=1$ –  Vahid Dec 18 '11 at 18:38
    
Your $f$ is simple. Take the $y$-axis pointing up. Stare at the $x,y,z$ system so that the plane appears as a line and the $y$ axis is pointing up. You are trying to find the height of the point of intersection of a line with a circle. –  David Mitra Dec 18 '11 at 18:54

2 Answers 2

I prefer to maximize $z$, the visualization is more familiar, since $z$ is "up."

Think about how the plane $x+y+z=1$ meets the sphere $x^2+y^2+z^2=3$. Note that the plane has $x$-$y$ symmetry.

The highest point on the circle of intersection has $x=y$. So at the maximum we have $z^2+2x^2=3$, $z+2x=1$. Solve for $z$. We get $z=5/3$.

Added: The geometry is clear, but here is some algebraic backup. We have $x^2+y^2=3-z^2$ and $x+y=1-z$. Thus
$$(x-y)^2=2(x^2+y^2)-(x+y)^2=(6-2z^2)-(1-2z+z^2)=5+2z-3z^2.$$ Since $5+2z-3z^2=(5-3z)(1+z)$, and $(x-y)^2 \ge 0$, we conclude that $$(5-3z)(1+z)\ge 0.$$ We are interested in positive $z$, so $5-3z \ge 0$, meaning that $z\le 5/3$, with equality when $x=y$.

Comment: About how to solve "this kind of problem," it depends how widely one interprets "this kind." The only fairly general kinds of techniques are numerical. For certain restricted classes of problems, Lagrange Multipliers can be useful. Here we took advantage of the symmetry. Special, perhaps, but many physically important problems have natural symmetries.

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You might try using Lagrange multipliers. Alternatively, note that the intersection of a sphere with a plane (when nontrivial) is a circle. Find a parametric representation of it. It might help to note that the vectors $(1,1,1)$, $(1,0,-1)$ and $(1,-2,1)$ are orthogonal.

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