<s>[INST] Proving inequality using Lagrange multipliers. I started learing about Lagrange Multipliers and I got the following question:
Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}≥\frac{3}{2}$, for each $a,b,c>0$.
I'm not sure how to use Lagrange multipliers for it... any ideas?
 [/INST] Let the objective function be 
$$f = a(c+a)(a+b) + b(b+c)(a+b) + c(b+c)(c+a)$$
and the constraint be 
$$g = (a+b)(b+c)(c+a) - k.$$
Note that if $g=0$, then 
$$\frac{f}{k} = \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}.$$
Solving $\nabla f-\lambda g = 0$ and $g=0$ we find $f$ is minimized for 
$a=b=c=k^{1/3}/2$, with value $3k/2$. 
Thus, 
$f/k \ge 3/2$, 
which proves the inequality. 
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