Tim Max $(3\sqrt{2}y-\sqrt{11}z)^2+(\sqrt{7}z-3\sqrt{2}x)^2+(\sqrt{11}x-\sqrt{7}y)^2$
 

Cho $x^2+y^2+x^2=1.$ Tim Max $(3\sqrt{2}y-\sqrt{11}z)^2+(\sqrt{7}z-3\sqrt{2}x)^2+(\sqrt{11}x-\sqrt{7}y)^2$

Lagrange Identity:

$\displaystyle (bc'-b'c)^2+(ca'-a'c)^2+(ab'-a'b)^2=(a^2+b^2+c^2)(a'^2+b'^2+c'^2)-(aa'+bb'+cc')^2$

$(3\sqrt{2}y-\sqrt{11}z)^2+(\sqrt{7}z-3\sqrt{2}x)^2+(\sqrt{11}x-\sqrt{7}y)^2=(18+11+7)-(\sqrt{7}x+\sqrt{11}y+3\sqrt{2}z)^2\leq 36.$

Dấu bằng xảy ra khi $(\sqrt{7}x+\sqrt{11}y+3\sqrt{2}z)=0$