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I am looking for the the supremum of the expression $x+y+z$, all real numbers, under the constraints $x^2+y^2=1$ and $z\leq y\leq x$. Thanks in advance for the help!

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HINT: You can take $z$ as big as you can till $z=y$ and it doesn't affect the constraint , so your problem reduces to maximize $x+2y$ given $x^2+y^2=1$ and $x\geq y$.

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So writing $x=r.cos(u)$ and $y=r.sin(u)$ we are done to maximizing $cos(u)+2.sin(u)$...am I right? –  Tina Sep 23 '12 at 19:16
    
another constraint is there, $\tan(u)\geq 1\implies \pi/4\leq u\leq \pi/2$ –  Aang Sep 23 '12 at 19:19
    
Right, so this gives us $u=Arctan(2)$, hence the maximum should be $cos(Arctan(2))+2.sin(Arctan(2))$...? –  Tina Sep 23 '12 at 20:33
    
@Tina: Since $\pi/4\leq\arctan(2)\leq\pi/2$, you got it right. –  Aang Sep 24 '12 at 5:39
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