Let $f(x,y,z):=z^4(x^2 −xy+y^2)+z^2(x^4 +y^4)$ and let $A ⊂ \mathbb{R}^3$ be the convex polyhedron with vertices (1, 0, 0), (0, 2, 0), (0, 0, 3), and (4, 4, 0). Find $sup_A f$.
I understand how to use lagrange multipliers on restraints and boundaries and have to check if jacobian matrix includes 0 or not correct? but I am not sure how to apply here. Thanks.