<s>[INST] Minimum and maximum value of function $f(x,y,z) = x^3 + y^3 + z^3$ Find minimum and maximum value of function $$f(x,y,z) = x^3 + y^3 + z^3$$ on set $$ \left\{ (x,y,z):  x^2 + y^2 + z^2 = 1  \wedge x+y+z =  \sqrt{3} \right\} $$
I don't know what is this set. We have sphere and plane so I suppose that it may be circle or point. How find it?
 [/INST] note 
$$(x^2+y^2+z^2)(1+1+1)\ge (x+y+z)^2$$
so if
$$x^2+y^2+z^2=1,x+y+z=\sqrt{3}$$
then we have
$$3(x^2+y^2+z^2)=(x+y+z)^2$$
$$\Longrightarrow x^2+y^2+z^2-xy-yz-xz=0$$
so 
$x=y=z$
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