Algebraic maximum and minimum based on a constraint Suppose $a,b,c$ are real numbers such that $a^2b^2 + b^2c^2 + c^2a^2 = k$, where $k$ is a constant. 
Then the set of all possible values of $abc(a+b+c)$ is?
I attempted writing the constraint in the form of $(abc)^2(1/a + 1/b + 1/c)$ and then tried using $(a+b+c)(1/a+1/b+1/c) >= 9$, but I think I'm missing something

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*It is known that $x^2+y^2+z^2\geq xy+yz+zx$. Note $x=ab,y=bc,z=ca, A=abc(a+b+c)$ it follows that $(ab)^2+(bc)^2+(ca)^2 \geq abc(a+b+c)<=>k\geq A.$

*$\dfrac{k}{2}+A = \dfrac{1}{2}[(ab)^2+(bc)^2+(ca)^2]+ abc(a+b+c)= \dfrac{1}{2}[(ab+bc)^2+(bc+ca)^2+(ca+ab)^2] \geq 0=>A\geq -\dfrac{k}{2}.$

