finding maximum of a function on a closed set 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 .
 A: 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.
A: 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.
