Since ab + bc + ac = 3, let ab, bc, and ac all equal 1. Now, 1³ + 1³ + 1³ + 6(1) = 9.

Is this the lower bound? My intuition says yes, because to decrease one of them, you'd have to increase at least one other, possible more than you decreased the one. However, I don't know how to prove it. Maybe someone else could step in?

LOL!! Gee, and I really feel like doing that with proofs ...


OK, let's just play with this, start with a=b=c=1, and then increase "a" by ε, and decrease "c" by λ to compensate

ab + bc + ac = (1+ε)1 + 1(1-λ) + (1+ε)(1-λ) = 1 + ε + 1 - λ + (1 - λ + ε - ελ)
= 3 + 2ε - 2λ - ελ = 3

∴ 2ε - 2λ - ελ = 0

So, does that "- ελ" term mean that ε has to be slightly larger than λ to make it work? And hence "a³ + b³ + c³ + 6abc" is going to increase?

I think so ...

... and I will now hypnotize you all ...