I have question about hessian matrices

Question

let f(x₁,x₂,x₃)= x₁+ x₂² + x₃³ - x₁x₂ -3x_{3 ,} (x₁,x₂,x₃) € R³

determine if f(x) is covex or cocave

the first x₂ is to the power 2 and first x₃ is to the power 3

but I couldn’t do that using the keyboard

Answer 1

I know the Hessian matrix would be [0 -1 0; -1 2 0; 0 0 6*z] ... normally you'd determine the signs of the eigenvalues to determine concavity, but this has a mixture of positive and negative so it should be an indefinite matrix, which should imply a saddle point at the stationary point