max. value

[man111]

If $x,y,z \in \mathbb{R}$ and $x+y+z=0,x^2+y^2+z^2=6$.Then Max.$(x^2y+y^2z+z^2x)$

[ИСН]

Let's be unreasonable.
Listen, man, you take a sphere and cut it open with a plane. You get a circle. A circle can be parametrized, say, like

$$\left\{\begin{array}{l} x=2\cos t, \\ y=2\cos \left({2\pi\over3}+t\right), \\ z=2\cos \left({4\pi\over3}+t\right), \end{array}\right.\,t\in[0,2\pi)$$

Which brings the whole thing down to old good 1D.

[arqady]

man111 в сообщении #419831 писал(а):

If $x,y,z \in \mathbb{R}$ and $x+y+z=0,x^2+y^2+z^2=6$.Then Max.$(x^2y+y^2z+z^2x)$

Докажите, что максимум равен $6$.