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 $x\geq y$.