<s>[INST] lagrange multiplier inequality question 
Prove for every $x,y,z>0, that\
f(x,y,z)= \frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}\geq \frac{3}{2} $

So since $f$ is a homogeneous function we can prove this on the following constraint $g(x,y,z)=x+y+z-1=0$.
However, I seem to be stuck now with this equation set:
$\frac{1}{y+z}-\frac{y}{(x+z)^2}-\frac{z}{(y+x)^2}= \lambda$
$\frac{1}{x+z}-\frac{x}{(y+z)^2}-\frac{z}{(y+x)^2}= \lambda$
$\frac{1}{x+y}-\frac{x}{(y+z)^2}-\frac{y}{(z+x)^2}= \lambda$
any help?
 [/INST] $$\frac{x}{y+z}=\frac{x+y+z}{y+z}-1$$
So we get $$LHS=\frac{2(x+y+z)(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x})}{2}-3 \ge \frac{9}{2}-3$$
Using AM-GM inequality,
$$LHS \ge \frac{9}{2}-3=\frac{3}{2}$$
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