Para$x,y,z>0$$xy+yz+xz=1$, minimizar $$P=\frac{1}{4x^{2}-yz+2}+\frac{1}{4y^{2}-zx+2}+\frac{1}{4z^{2}-xy+2}$$
Yo: Vamos a $xy=a; yz=b;zx=c \Rightarrow a+b+c=1$
$\Rightarrow x^2=\frac{ac}{b};y^2=\frac{ab}{c};z^2=\frac{bc}{a}$
Por lo tanto $$P=\sum \frac{1}{\frac{4ac}{b}-b+2}=\sum \frac{1}{\frac{4ac}{b}-b+2(a+b+c)}=\sum \frac{1}{\frac{4ac}{b}+2a+b+2c}=\sum \frac{b}{4ac+2ab+b^2+2bc}$$
$$P=\sum \frac{b}{(2a+b)(2c+b)} \geq \sum \frac{4b}{(2a+2b+2c)^2}=\sum \frac{b}{(a+b+c)^2}=1$$
Y necesito nuevo método ?