<s>[INST] Inequality with a, b, c about finding minimal and maximal value Find the minimal and maximal value (if they exist) of ${\sqrt{\frac{a(b+c)}{b^2+c^2}}} +{\sqrt{\frac{b(a+c)}{a^2+c^2}}} +{\sqrt{\frac{c(b+a)}{b^2+a^2}}}$ if  are non-negative real numbers, such that at least two of them are positive.
My attempts for the case where all variables are positive:
I tried applying AM-GM on ${\sqrt{a(b+c)}} $ and etc. and then applying $b^2+c^2=>(b+c)^2/2$,but the inequality I received was false. I also tried rewriting $a(b+c)/(b^2+c^2)=(b/a+c/a)/((b/a)^2+(c/a)^2)$ and etc. and then letting $a/b=x$, $b/c=y$ and $c/a=z$ and etc but I was stuck from there. I also tried applying Cauchy Schwarz by squaring the whole expression.
 [/INST] For $c=0$ and $a=b=1$ we obtain a value $2$.
We'll prove that it's a minimal value.
Indeed, by Holder:
$$\left(\sum_{cyc}\sqrt{\frac{a(b+c)}{b^2+c^2}}\right)^2\sum_{cyc}\frac{a^2(b^2+c^2)}{b+c}\geq(a+b+c)^3.$$
Thus, it's enough to prove that:
$$(a+b+c)^3\geq4\sum_{cyc}\frac{a^2(b^2+c^2)}{b+c},$$ which is obvious after full expanding:
$$\sum_{sym}(a^5b-a^3b^3+5a^4bc+11a^3b^2c+5a^2b^2c^2)\geq0.$$
The maximal value does not exist
because for $b=c=1$ and $a\rightarrow+\infty$ our expression close to $+\infty$.
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