$A= x^3 + y^3 +z^3 + kxyz $

Question

Let $x,y,z$ are three sides of a triangle such that $x+y+z=3$. Find $k \geq 0$ such that: $$A= x^3 + y^3 +z^3 + kxyz $$ have minimum.

I tried for this: $k=6,k=15,k=\frac{15}{4}$ and it have all minimum, which is easy with Schur.