<s>[INST] Find the maximum and minimum value of $P =x+y+z+xy+yz+zx$ Let $x^2+y^2+z^2\leq27$ and $P = x+y+z+xy+yz+zx$. Find the value of $x, y, z$ such that $P$ is the maximum value and minimum value.
My attempt :
$$(x-y)^2 + (y-z)^2 + (z-x)^2 \geq 0$$
$$27 \geq x^2+y^2+z^2 \geq xy+yz+zx\tag{1}$$
$$(x+y+z)^2 \leq 3(x^2+y^2+z^2) \le 3 \cdot 27$$
$$(x+y+z)^2 \leq 81$$
$$x+y+z \leq 9\tag{2}$$
From $(1), (2)$, $ x+y+z+xy+yz+zx \leq 36$, so $P_{\text{max}} = 36$ with equality hold at $x=y=z=3$.
Please suggest how to find $P_{\text{min}}$.
 [/INST] A minimum is $-14$.
Prove that $$(x+y+z)\sqrt{\frac{x^2+y^2+z^2}{27}}+xy+xz+yz+\frac{14}{27}(x^2+y^2+z^2)\geq0$$
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