<s>[INST] Inequality - Maximize k such that I'm not sure how to solve this problem I came across online. Here it is:

Let $a, b, c$ be nonnegative reals. Maximize $k$ such that 
  $$(a+b+c)\left(\dfrac 1a + \dfrac 1b + \dfrac 1c\right) + \dfrac{k(ab + bc+ca)}{a^2+b^2+c^2} \geq 9 + k.$$

What I know is this:
$a, b, c$ must be positive, because otherwise we'd have division by zero. I also know that $(a+b+c)\left(\dfrac 1a + \dfrac 1b + \dfrac 1c\right) \geq 9$ for positive $a,b,c$. 
 [/INST] Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.
Hence, our inequality it's $f(w^3)\geq0$, where $f$ is a decreasing function.
Thus, it remains to prove our inequality for a maximal value of $w^3$.
We know that $a$, $b$ and $c$ are positive roots of the following equation
$$(x-a)(x-b)(x-c)=0$$ or
$$x^3-3ux^2+3v^2x-w^3=0$$ or
$$x^3-3ux^2+3v^2x=w^3,$$
which says that a graph of $f(x)=x^3-3ux^2+3v^2x$ and the line $y=w^3$ 
have three common points: $(a,f(a))$, $(b,f(b))$ and $(c,f(c))$. 
Now, let $a\leq b\leq c$ and since $u^2\geq v^2$, we see that $$f'(x)=3x^2-6xu+3v^2=3(x^2-2ux+v^2)=3(x-x_1)(x-x_2).$$
Let $x_1\leq x_2$.
We see that $x_1>0$ and $\left(x_1,f(x_1)\right)$ is a maximum point and $\left(x_2,f(x_2)\right)$ is a minimum point.
Now, let $w^3$ increases and $u$ and $v^2$ are constants.
Thus, $a$, $b$ and $c$ are change and $w^3$ will get a maximal value, 
when the line $y=w^3$ will touch to the graph of $f$, 
which happens for equality case of two variables (for $a=b$ in our case). 
Since our inequality is homogeneous, we can assume $b=c=1$, which gives
$$(a+2)\left(\frac{1}{a}+2\right)-9\geq k\left(1-\frac{2a+1}{a^2+2}\right)$$ or
$$\frac{2(a^2+2)}{a}\geq k.$$
Now, by AM-GM
$$\frac{2(a^2+2)}{a}=2\left(a+\frac{2}{a}\right)\geq2\cdot2\sqrt2=4\sqrt2,$$
which gives the answer $4\sqrt2$.
Done!
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