<s>[INST] Finding an interval containing the values of $ abc(a + b + c) $, given a quadratic constraint Suppose $ a, b, c $ are real numbers such that $$ (ab)^2 + (bc)^2 + (ca)^2 = k $$
then all possible values of $ abc(a + b + c) $ lies between which interval?
 [/INST] Note $$x^2+y^2+z^2\ge xy+yz+zx$$
then we have
$$k=(ab)^2+(bc)^2+(ac)^2\ge abc(a+b+c)$$
 other hand 
$$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+xz)\ge 0$$
so
$$xy+yz+xz\ge -\dfrac{1}{2}(x^2+y^2+z^2)$$
so
$$abc(a+b+c)\ge -\dfrac{1}{2}[(ab)^2+(bc)^2+(ac)^2]=-\dfrac{k}{2}$$
so
$$abc(a+b+c)\in [-\dfrac{k}{2},k]$$
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