<s>[INST] $a,b,c\in\Bbb R^+, x,y,z\in \Bbb R, $ show that $\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c} \ge \frac{(x+y+z)^2}{a+b+c}$ $a,b,c\in\Bbb R^+, x,y,z\in \Bbb R, $ show that $\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c} \ge \frac{(x+y+z)^2}{a+b+c}$ (use Cauchy–Schwarz inequality)
I have trouble finding the two vectors. Is it $(x,y,z)$ and $(a,b,c)$?
I need a hint
 [/INST] Hint: Use Cauchy Schwarz in Engelform:
$$\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}\geq \frac{(x+y+z)^2}{a+b+c}$$
It is equivalent to $${z}^{2}{a}^{2}b+{y}^{2}{a}^{2}c+{z}^{2}a{b}^{2}-2\,abcxy-2\,abcxz-2\,a
bcyz+{y}^{2}a{c}^{2}+{x}^{2}{b}^{2}c+{x}^{2}b{c}^{2}
\geq 0$$
and this is $$a(bz-cy)^2+b(az-xc)^2+c(bx-ay)^2\geq 0$$ if $$a,b,c$$ are positive.
</s>